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<!DOCTYPE html>
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Stats key points - HackMD
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<div id="doc" class="markdown-body container-fluid"><h1 id="stats-key-points"><a class="anchor hidden-xs" href="#stats-key-points" title="stats-key-points"><span class="octicon octicon-link"></span></a>Stats key points</h1><h2 id="ss02"><a class="anchor hidden-xs" href="#ss02" title="ss02"><span class="octicon octicon-link"></span></a>SS02</h2><h3 id="time-series-analysis"><a class="anchor hidden-xs" href="#time-series-analysis" title="time-series-analysis"><span class="octicon octicon-link"></span></a>Time series analysis</h3><h4 id="seasonal-variation"><a class="anchor hidden-xs" href="#seasonal-variation" title="seasonal-variation"><span class="octicon octicon-link"></span></a>Seasonal variation</h4><p>When asked for the type of variation</p><p><strong>Seasonal variation</strong> If the variations from the fit line appear to follow a pattern</p><p><strong>Random variation</strong> If they don’t</p><h4 id="moving-averages-and-seasonal-effects"><a class="anchor hidden-xs" href="#moving-averages-and-seasonal-effects" title="moving-averages-and-seasonal-effects"><span class="octicon octicon-link"></span></a>Moving averages and seasonal effects</h4><p><strong>Moving averages</strong> For <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-214-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>x</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="false" stretchy="false">}</mo><mo>,</mo><mtext>&#xA0;</mtext><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mn>3</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2680" style="width: 17.574em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.398em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.437em 1014.4em 3.128em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2681"><span class="mo" id="MathJax-Span-2682" style="font-family: STIXGeneral;">{</span><span class="msubsup" id="MathJax-Span-2683"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2684" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-2685" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; 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margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.435em 1003.69em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -1.842em;"><span class="mrow" id="MathJax-Span-2710"><span class="msubsup" id="MathJax-Span-2711"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2712" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.31em;"><span class="texatom" id="MathJax-Span-2713"><span class="mrow" id="MathJax-Span-2714"><span class="mi" id="MathJax-Span-2715" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">i</span><span class="mo" id="MathJax-Span-2716" style="font-size: 50%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2717" style="font-size: 50%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2718" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="msubsup" id="MathJax-Span-2719"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2720" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.31em;"><span class="mi" id="MathJax-Span-2721" style="font-size: 50%; font-family: STIXGeneral; 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font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-2729" style="font-size: 50%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-2730" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.79em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.794em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.559em; border-left: 0px solid; width: 0px; height: 1.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><msub><mi>x</mi><mn>3</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo fence="false" stretchy="false">}</mo><mo>,</mo><mtext> </mtext><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mn>3</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-214">\{x_1, x_2, x_3, ..., x_n\},\ a_i = \frac{x_{i-1} + x_i + x_{i+1}}{3}</script></span></p><p><strong>Centered moving average</strong> The average of the current moving average and the surrounding moving averages</p><p>The seasonal effect is the average of the values of the difference between the actual values and the centered moving average.</p><p>Take the centered moving average and subtract it from the actual value for each item in a group. Then calculate the average of these values.</p><h4 id="estimation-from-seasonal-effects"><a class="anchor hidden-xs" href="#estimation-from-seasonal-effects" title="estimation-from-seasonal-effects"><span class="octicon octicon-link"></span></a>Estimation from seasonal effects</h4><p>Once the seasonal effects has been calculated the fit line can be used to find a value for a given time, and the seasonal effect can then be applied to this value to produce an estimate</p><h3 id="sampling"><a class="anchor hidden-xs" href="#sampling" title="sampling"><span class="octicon octicon-link"></span></a>Sampling</h3><h4 id="simple-random-samples"><a class="anchor hidden-xs" href="#simple-random-samples" title="simple-random-samples"><span class="octicon octicon-link"></span></a>Simple random samples</h4><ol>
<li>Assign a range of values to the data</li>
<li>Choose random values from the number table, starting from a random position</li>
</ol><p>Each item has the same probability of being chosen.</p><p>If the data is in sections, each section may not be represented.<br>
May not give correct representations of strata.</p><h4 id="stratified-random-sampling"><a class="anchor hidden-xs" href="#stratified-random-sampling" title="stratified-random-sampling"><span class="octicon octicon-link"></span></a>Stratified random sampling</h4><p>There may often be factors which divide up the population into groups (strata), and we may expect the measurement of interest to vary among the different groups.<br>
This can be accounted for when we select a sample from the population in order that we obtain a sample that is representative of the population.</p><p>We generally require that the proportion of each stratum in the sample should be the same as in the population.</p><p>Stratified sampling techniques are generally used when the population is heterogeneous, or dissimilar, where certain homogeneous sub populations can be isolated.</p><p>Some reasons for using stratified sampling over simple random sampling are:</p><ul>
<li>The cost per observation may be reduced</li>
<li>Estimates of the population parameters may be wanted for each sub-population</li>
<li>Increased accuracy at a given cost</li>
</ul><h4 id="cluster-quota-and-systematic-sampling"><a class="anchor hidden-xs" href="#cluster-quota-and-systematic-sampling" title="cluster-quota-and-systematic-sampling"><span class="octicon octicon-link"></span></a>Cluster, quota, and systematic sampling</h4><h5 id="cluster-sampling"><a class="anchor hidden-xs" href="#cluster-sampling" title="cluster-sampling"><span class="octicon octicon-link"></span></a>Cluster sampling</h5><p>The data is is divided into separate groups. Then a simple random sample of clusters is selected from the population.</p><p>The population is divided into <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-215-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2731" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2732"><span class="mi" id="MathJax-Span-2733" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-215">N</script></span> groups, called clusters</p><p>The researcher randomly selects <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-216-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2734" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2735"><span class="mi" id="MathJax-Span-2736" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-216">n</script></span> clusters to include in the sample</p><p>The number of observations within each cluster <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-217-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>j</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2737" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.666em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2738"><span class="msubsup" id="MathJax-Span-2739"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.87em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2740" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.822em;"><span class="mi" id="MathJax-Span-2741" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>j</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-217">M_j</script></span> is known, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-218-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mo>&#x2211;</mo><msub><mi>M</mi><mi>j</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2742" style="width: 5.433em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.46em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em 1004.46em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2743"><span class="mi" id="MathJax-Span-2744" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2745" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-2746" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.31em;">∑</span><span class="msubsup" id="MathJax-Span-2747" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.87em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2748" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.822em;"><span class="mi" id="MathJax-Span-2749" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.441em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mo>∑</mo><msub><mi>M</mi><mi>j</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-218">M = \sum M_j</script></span>.</p><p><strong>One-stage sampling</strong> All of the elements within selected clusters are included in the sample</p><p><strong>Two-stage sampling</strong> A subset of elements within selected clusters are randomly selected for inclusion in the sample</p><p>Sometimes, the cost per sample point is less for cluster sampling than for other sampling methods. Given a fixed budget, the researcher may be able to use a bigger sample with cluster sampling than with the other methods. When the increased sample size is sufficient to offset the loss in precision, cluster sampling may be the best choice.</p><p>Application:</p><ol>
<li>Select a cluster grouping as a sampling frame</li>
<li>Mark each cluster with a unique number</li>
<li>Choose a sample of clusters applying probability sampling</li>
</ol><h4 id="quota-sampling"><a class="anchor hidden-xs" href="#quota-sampling" title="quota-sampling"><span class="octicon octicon-link"></span></a>Quota sampling</h4><p>Quota sampling requires, that representative individuals are chosen from a specific subgroup</p><p>Advantages:</p><ul>
<li>Primary collection can be done in a short time</li>
<li>The application of quota sampling can save costs and time</li>
<li>Quota sampling is not dependent on the presence of the sampling frames.</li>
</ul><p>Disadvantages:</p><ul>
<li>It is not possible to calculate the sampling error</li>
<li>Other important characteristics may be disproportionately represented in the final sample group</li>
<li>There is a potential for researcher bis and the quality of work may suffer due to researcher incompetency</li>
</ul><h4 id="systematic-sampling"><a class="anchor hidden-xs" href="#systematic-sampling" title="systematic-sampling"><span class="octicon octicon-link"></span></a>Systematic sampling</h4><p>A sampling method in which the first position in the data set is randomly chosen, and every <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-219-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mi>h</mi></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2750" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.13em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2751"><span class="msubsup" id="MathJax-Span-2752"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2753" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="texatom" id="MathJax-Span-2754"><span class="mrow" id="MathJax-Span-2755"><span class="mi" id="MathJax-Span-2756" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2757" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">h</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mi>h</mi></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-219">n^{th}</script></span> position after this is also chosen</p><p>Advantages:</p><ul>
<li>Allows the researcher to add a degree of system or process into the random selection</li>
<li>Known and equal probability of selection</li>
<li>The assurance that the population will be evenly sampled. Simple random sampling allows a clustered selection of subjects</li>
</ul><p>Disadvantages:</p><ul>
<li>The process of selection can interact with a hidden periodic trait within the population</li>
<li>The process can also hide a periodic trait</li>
</ul><h3 id="discrete-probability-distributions"><a class="anchor hidden-xs" href="#discrete-probability-distributions" title="discrete-probability-distributions"><span class="octicon octicon-link"></span></a>Discrete probability distributions</h3><h4 id="expectations-and-variance"><a class="anchor hidden-xs" href="#expectations-and-variance" title="expectations-and-variance"><span class="octicon octicon-link"></span></a>Expectations and variance</h4><h4 id="poisson"><a class="anchor hidden-xs" href="#poisson" title="poisson"><span class="octicon octicon-link"></span></a>Poisson</h4><p>(Same as MS02)</p><h3 id="interpretation-of-data"><a class="anchor hidden-xs" href="#interpretation-of-data" title="interpretation-of-data"><span class="octicon octicon-link"></span></a>Interpretation of data</h3><h4 id="pie-charts"><a class="anchor hidden-xs" href="#pie-charts" title="pie-charts"><span class="octicon octicon-link"></span></a>Pie charts</h4><h4 id="line-diagrams"><a class="anchor hidden-xs" href="#line-diagrams" title="line-diagrams"><span class="octicon octicon-link"></span></a>Line diagrams</h4><h4 id="box-and-whisker-plots"><a class="anchor hidden-xs" href="#box-and-whisker-plots" title="box-and-whisker-plots"><span class="octicon octicon-link"></span></a>Box and whisker plots</h4><h4 id="frequency-diagrams"><a class="anchor hidden-xs" href="#frequency-diagrams" title="frequency-diagrams"><span class="octicon octicon-link"></span></a>Frequency diagrams</h4><h4 id="scatter-diagrams"><a class="anchor hidden-xs" href="#scatter-diagrams" title="scatter-diagrams"><span class="octicon octicon-link"></span></a>Scatter diagrams</h4><p><strong>Histograms not required</strong></p><h3 id="hypothesis-testing"><a class="anchor hidden-xs" href="#hypothesis-testing" title="hypothesis-testing"><span class="octicon octicon-link"></span></a>Hypothesis testing</h3><h4 id="tests-for-means"><a class="anchor hidden-xs" href="#tests-for-means" title="tests-for-means"><span class="octicon octicon-link"></span></a>Tests for means</h4><h4 id="errors"><a class="anchor hidden-xs" href="#errors" title="errors"><span class="octicon octicon-link"></span></a>Errors</h4><p><strong>Type I error</strong>: Rejecting a true null hypothesis</p><p><strong>Type II error</strong>: Accepting a false null hypothesis</p><h1 id="if-it-says-test-state-the-fucking-hypotheses"><a class="anchor hidden-xs" href="#if-it-says-test-state-the-fucking-hypotheses" title="if-it-says-test-state-the-fucking-hypotheses"><span class="octicon octicon-link"></span></a>If it says test. State the fucking hypotheses!</h1><h1 id="tests"><a class="anchor hidden-xs" href="#tests" title="tests"><span class="octicon octicon-link"></span></a>Tests</h1><table>
<thead>
<tr>
<th>Test</th>
<th>Details</th>
<th>Case to reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="msubsup" id="MathJax-Span-3"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-4" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-5" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-1">H_0</script></span></th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>SS02</strong></td>
<td></td>
<td></td>
</tr>
<tr>
<td>Z/T test</td>
<td>Determines whether two population means are different</td>
<td>TV > CV</td>
</tr>
<tr>
<td><strong>SS03</strong></td>
<td></td>
<td></td>
</tr>
<tr>
<td>Contingency tables</td>
<td>A test for independence</td>
<td>CV > TV</td>
</tr>
<tr>
<td>Sign test</td>
<td>Test for difference in medians</td>
<td>TV < CV</td>
</tr>
<tr>
<td>Wilcoxon</td>
<td>Test for difference in mean or median</td>
<td>TV < CV</td>
</tr>
<tr>
<td>Mann-Whitney</td>
<td>Test for equality of population</td>
<td>TV < CV</td>
</tr>
<tr>
<td>Kruskal-Wallis</td>
<td>Test for equality of population of two or more samples</td>
<td>TV > CV</td>
</tr>
<tr>
<td>Correlation coefficient</td>
<td>Test for existence of correlation between two random variables</td>
<td>TV > SL</td>
</tr>
<tr>
<td><strong>SS04</strong></td>
<td></td>
<td></td>
</tr>
<tr>
<td>Poisson</td>
<td>Test for change in a Poisson variable</td>
<td>TV < SL</td>
</tr>
<tr>
<td>Proportion</td>
<td>Test whether sample proportion represents the population</td>
<td>TV < SL</td>
</tr>
<tr>
<td><strong>SS05</strong></td>
<td></td>
<td></td>
</tr>
<tr>
<td>Variance</td>
<td>Tests a sample for a given population variance</td>
<td>TV > CV(Upper) or TV < CV(Lower)</td>
</tr>
<tr>
<td>Variance equality of samples (F)</td>
<td>Test for equality of the variances of the populations of two samples of two normally distributed random variables</td>
<td>TV > CV</td>
</tr>
<tr>
<td>Difference in mean (Two sample Z)</td>
<td>Test for the difference in the means of two independent populations</td>
<td>|TV| > |CV|</td>
</tr>
<tr>
<td>Difference in mean (Two sample T)</td>
<td>Test for the difference in the means of two independent populations with unknown variances</td>
<td>|TV| > |CV|</td>
</tr>
<tr>
<td>Goodness of fit</td>
<td>Test for the fit of a sample to a particular distributio n</td>
<td>TV > CV</td>
</tr>
<tr>
<td><strong>SS06</strong></td>
<td></td>
<td></td>
</tr>
<tr>
<td>Paired comparisons</td>
<td>Analysis of the difference between pairs of values sampled from two normal populations</td>
<td>TV > CV</td>
</tr>
<tr>
<td>Analysis of variance</td>
<td>An extension of F tests with more than 2 populations</td>
<td>TV > CV</td>
</tr>
<tr>
<td>Two way analysis of variance</td>
<td>An analysis of variance which accounts for a second factor</td>
<td>TV > CV</td>
</tr>
<tr>
<td>Latin squares</td>
<td></td>
<td></td>
</tr>
</tbody>
</table><h1 id="ss03"><a class="anchor hidden-xs" href="#ss03" title="ss03"><span class="octicon octicon-link"></span></a>SS03</h1><h2 id="contingency-tables"><a class="anchor hidden-xs" href="#contingency-tables" title="contingency-tables"><span class="octicon octicon-link"></span></a>Contingency tables</h2><ol>
<li>State hypotheses (In context)<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-7"><span class="msubsup" id="MathJax-Span-8"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-9" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-10" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-2">H_0</script></span>: No assocation<br>
ii <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-11" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-12"><span class="msubsup" id="MathJax-Span-13"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-14" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-15" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-3">H_1</script></span>: Assocation</li>
<li>Calculate the expected values<br>
If any value is less than 5, merge the rows or columns</li>
<li>Calculate the text <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C7;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-16" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1000.98em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-17"><span class="msubsup" id="MathJax-Span-18"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-19" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-20" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-4">\chi^2</script></span> statistic</li>
<li>Find the critical value for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BD;</mi><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>m</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-21" style="width: 9.48em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.738em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1007.69em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-22"><span class="mi" id="MathJax-Span-23" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-24" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-25" style="font-family: STIXGeneral; padding-left: 0.31em;">(</span><span class="mi" id="MathJax-Span-26" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-27" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-28" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-29" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-30" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-31" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mo" id="MathJax-Span-32" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-33" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-34" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ν</mi><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-5">\nu = (n-1)(m-1)</script></span> degrees of freedom<br>
This can either be done from the table or with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CHISQUARE_ICDF</mtext><mo stretchy="false">(</mo><mi>&#x03BD;</mi><mo>,</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-35" style="width: 15.013em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.298em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1012.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-36"><span class="mtext" id="MathJax-Span-37" style="font-family: STIXGeneral;">CHISQUARE_ICDF</span><span class="mo" id="MathJax-Span-38" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-39" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-40" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-41" style="font-family: STIXGeneral; padding-left: 0.207em;">1</span><span class="mo" id="MathJax-Span-42" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-43" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-44" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CHISQUARE_ICDF</mtext><mo stretchy="false">(</mo><mi>ν</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-6">\text{CHISQUARE_ICDF}(\nu, 1-\alpha)</script></span></li>
<li>If text <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>&#x03C7;</mi><mi>t</mi><mn>2</mn></msubsup><mo>&gt;</mo><msubsup><mi>&#x03C7;</mi><mi>c</mi><mn>2</mn></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-45" style="width: 3.947em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.23em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1003.23em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-46"><span class="msubsup" id="MathJax-Span-47"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-48" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-49" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.31em 4.152em -999.997em); top: -3.788em; left: 0.464em;"><span class="mi" id="MathJax-Span-50" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-51" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="msubsup" id="MathJax-Span-52" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-53" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-54" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-55" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.441em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>χ</mi><mi>t</mi><mn>2</mn></msubsup><mo>></mo><msubsup><mi>χ</mi><mi>c</mi><mn>2</mn></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-7">\chi^2_t > \chi^2_c</script></span>, reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-56" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-57"><span class="msubsup" id="MathJax-Span-58"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-59" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-60" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-8">H_0</script></span> in context</li>
</ol><h3 id="yates’-correction"><a class="anchor hidden-xs" href="#yates’-correction" title="yates’-correction"><span class="octicon octicon-link"></span></a>Yates’ correction</h3><p>For a 2 by 2 table, Yates’ correction is used.</p><p>Rather than <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>&#x03C7;</mi><mn>2</mn></msup><mo>=</mo><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-61" style="width: 10.146em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.302em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1008.3em 3.947em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-62"><span class="msubsup" id="MathJax-Span-63"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-64" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.566em;"><span class="mn" id="MathJax-Span-65" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-66" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-67" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mfrac" id="MathJax-Span-68" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 4.306em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1004.15em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -2.098em;"><span class="mrow" id="MathJax-Span-69"><span class="mo" id="MathJax-Span-70" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-71"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-72" style="font-family: STIXGeneral; font-style: italic;">O</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mi" id="MathJax-Span-73" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-74" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-75" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-76" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-77" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-78"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-79" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-80" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.87em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-81"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-82" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-83" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.31em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.306em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}</script></span> the corrected formula is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>&#x03C7;</mi><mn>2</mn></msup><mo>=</mo><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>O</mi><mi>i</mi></msub><mo>&#x2212;</mo><msub><mi>E</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>&#x2212;</mo><mn>0.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-84" style="width: 13.886em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.376em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1011.38em 3.947em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-85"><span class="msubsup" id="MathJax-Span-86"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-87" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.566em;"><span class="mn" id="MathJax-Span-88" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-89" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-90" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mfrac" id="MathJax-Span-91" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1007.28em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -3.635em;"><span class="mrow" id="MathJax-Span-92"><span class="mo" id="MathJax-Span-93" style="font-family: STIXGeneral;">(</span><span class="texatom" id="MathJax-Span-94"><span class="mrow" id="MathJax-Span-95"><span class="mo" id="MathJax-Span-96" style="font-family: STIXVariants;">|</span></span></span><span class="msubsup" id="MathJax-Span-97"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-98" style="font-family: STIXGeneral; font-style: italic;">O</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mi" id="MathJax-Span-99" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-100" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-101" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-102" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-103" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="texatom" id="MathJax-Span-104"><span class="mrow" id="MathJax-Span-105"><span class="mo" id="MathJax-Span-106" style="font-family: STIXVariants;">|</span></span></span><span class="mo" id="MathJax-Span-107" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-108" style="font-family: STIXGeneral; padding-left: 0.259em;">0.5</span><span class="msubsup" id="MathJax-Span-109"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-110" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-111" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.87em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-112"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-113" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-114" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1007.38em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.38em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>χ</mi><mn>2</mn></msup><mo>=</mo><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>O</mi><mi>i</mi></msub><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>−</mo><mn>0.5</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><msub><mi>E</mi><mi>i</mi></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-10">\chi^2 = \sum \frac{(|O_i-E_i|-0.5)^2}{E_i}</script></span></p><h2 id="distribution-free-methods"><a class="anchor hidden-xs" href="#distribution-free-methods" title="distribution-free-methods"><span class="octicon octicon-link"></span></a>Distribution free methods</h2><table>
<thead>
<tr>
<th>Test</th>
<th>For</th>
<th>Use case</th>
</tr>
</thead>
<tbody>
<tr>
<td>Sign test</td>
<td>Median</td>
<td>When Wilcoxon cannot be used as data is not symmetrical or is non-numeric</td>
</tr>
<tr>
<td>Wilcoxon signed-rank test</td>
<td>Median or Mean</td>
<td>When a z or t test cannot be used</td>
</tr>
<tr>
<td>Mann-Whitney U test</td>
<td>Equality of populations of two samples</td>
<td>There are two samples</td>
</tr>
<tr>
<td>Kruskal-Wallis test</td>
<td>Equality of populations of two or more samples</td>
<td>There are more than two samples</td>
</tr>
</tbody>
</table><h3 id="sign-test"><a class="anchor hidden-xs" href="#sign-test" title="sign-test"><span class="octicon octicon-link"></span></a>Sign test</h3><p>The sign test checks for a difference in the median value by comparing each pair. It does not require a symmetric distribution and could be used on non-numeric data so long as the data can be assigned to two groups (e.g. boolean values of opinions)</p><ol>
<li>State hypotheses<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-115" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-116"><span class="msubsup" id="MathJax-Span-117"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-118" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-119" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-11">H_0</script></span> that <strong>population</strong> medians are equal <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B7;</mi><mn>1</mn></msub><mo>=</mo><msub><mi>&#x03B7;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-120" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1003.13em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-121"><span class="msubsup" id="MathJax-Span-122"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-123" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-124" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-125" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-126" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-127" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-128" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>η</mi><mn>1</mn></msub><mo>=</mo><msub><mi>η</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-12">\eta_1 = \eta_2</script></span><br>
ii. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-129" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-130"><span class="msubsup" id="MathJax-Span-131"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-132" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-133" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-13">H_1</script></span>: that <strong>population</strong> medians are not equal <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B7;</mi><mn>1</mn></msub><mo>&#x2260;</mo><msub><mi>&#x03B7;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-134" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.13em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-135"><span class="msubsup" id="MathJax-Span-136"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-137" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-138" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-139" style="font-family: STIXGeneral; padding-left: 0.31em;">≠</span><span class="msubsup" id="MathJax-Span-140" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-141" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-142" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>η</mi><mn>1</mn></msub><mo>≠</mo><msub><mi>η</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-14">\eta_1 \neq \eta_2</script></span> or <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B7;</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mo>&gt;</mo><mtext>or</mtext><mo>&lt;</mo><mo stretchy="false">)</mo><msub><mi>&#x03B7;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-143" style="width: 6.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.33em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.33em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-144"><span class="msubsup" id="MathJax-Span-145"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-146" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-147" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-148" style="font-family: STIXGeneral;">(</span><span class="mo" id="MathJax-Span-149" style="font-family: STIXGeneral;">></span><span class="mtext" id="MathJax-Span-150" style="font-family: STIXGeneral; padding-left: 0.31em;">or</span><span class="mo" id="MathJax-Span-151" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mo" id="MathJax-Span-152" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-153"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-154" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-155" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>η</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mo>></mo><mtext>or</mtext><mo><</mo><mo stretchy="false">)</mo><msub><mi>η</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-15">\eta_1 (> \text{or} <) \eta_2</script></span></li>
<li>Find the differences between each pair, ignoring any equal values</li>
<li>Count <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-156" style="width: 1.591em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1001.28em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-157"><span class="msubsup" id="MathJax-Span-158"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-159" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-160" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-16">T^+</script></span> the number of positive differences and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&#x2212;</mo></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-161" style="width: 1.591em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1001.28em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-162"><span class="msubsup" id="MathJax-Span-163"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-164" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-165" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-17">T^-</script></span> the number of negative differences</li>
<li>Find the value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow><mo>(</mo><mrow><mi>X</mi><mo>&lt;</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>+</mo></msup><mo>,</mo><msup><mi>T</mi><mo>&#x2212;</mo></msup><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mtext>&#xA0;</mtext><mi>X</mi><mo>&#x223C;</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-166" style="width: 17.83em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.603em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1014.45em 3.486em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-167"><span class="mi" id="MathJax-Span-168" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mrow" id="MathJax-Span-169" style="padding-left: 0.207em;"><span class="mo" id="MathJax-Span-170" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mrow" id="MathJax-Span-171"><span class="mi" id="MathJax-Span-172" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-173" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-174" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-175" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-176" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-177" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-178"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-179" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-180" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-181" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-182" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-183" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-184" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-185" style="font-family: STIXGeneral;">)</span><span class="mtext" id="MathJax-Span-186" style="font-family: STIXGeneral;"> </span><span class="texatom" id="MathJax-Span-187"><span class="mrow" id="MathJax-Span-188"><span class="mo" id="MathJax-Span-189" style="font-family: STIXVariants;">|</span></span></span><span class="mtext" id="MathJax-Span-190" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-191" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-192" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-193" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">B</span><span class="mo" id="MathJax-Span-194" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-195" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-196" style="font-family: STIXGeneral;">,</span><span class="mfrac" id="MathJax-Span-197" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-198" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-199" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-200" style="font-family: STIXGeneral;">)</span></span><span class="mo" id="MathJax-Span-201" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow><mo>(</mo><mrow><mi>X</mi><mo><</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msup><mi>T</mi><mo>+</mo></msup><mo>,</mo><msup><mi>T</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mtext> </mtext><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-18">P\left(X < min(T^+, T^-)\ |\ X\sim B(n, \frac 1 2)\right)</script></span></li>
<li>Compare the value above with the significance level. If it is less, reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-202" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-203"><span class="msubsup" id="MathJax-Span-204"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-205" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-206" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-19">H_0</script></span></li>
</ol><h3 id="wilcoxon-signed-rank-test"><a class="anchor hidden-xs" href="#wilcoxon-signed-rank-test" title="wilcoxon-signed-rank-test"><span class="octicon octicon-link"></span></a>Wilcoxon Signed-rank test</h3><p>The Wilcoxon test is similar to the sign test except that it ranks the differences ignoring their signs.</p><ol>
<li>State the hypotheses<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-207" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-208"><span class="msubsup" id="MathJax-Span-209"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-210" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-211" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-20">H_0</script></span> <strong>Population</strong> average difference (in mean or median) of 0<br>
ii. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-212" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-213"><span class="msubsup" id="MathJax-Span-214"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-215" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-216" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-21">H_1</script></span> <strong>Population</strong> average difference not 0</li>
<li>Rank the absolutes of the differences, giving each rank the sign of its respective difference</li>
<li>Calculate <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-217" style="width: 1.591em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1001.28em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-218"><span class="msubsup" id="MathJax-Span-219"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-220" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-221" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>+</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-22">T^+</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>&#x2212;</mo></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-222" style="width: 1.591em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1001.28em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-223"><span class="msubsup" id="MathJax-Span-224"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-225" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-226" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>T</mi><mo>−</mo></msup></math></span></span><script type="math/tex" id="MathJax-Element-23">T^-</script></span>, the sums of the positive and negative ranks respectively. Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub><mo>=</mo><mtext>min</mtext><mo stretchy="false">(</mo><msup><mi>T</mi><mo>+</mo></msup><mo>,</mo><msup><mi>T</mi><mo>&#x2212;</mo></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-227" style="width: 9.019em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em 1007.33em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-228"><span class="msubsup" id="MathJax-Span-229"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-230" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-231" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-232" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mtext" id="MathJax-Span-233" style="font-family: STIXGeneral; padding-left: 0.31em;">min</span><span class="mo" id="MathJax-Span-234" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-235"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-236" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-237" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-238" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-239" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-240" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.72em;"><span class="mo" id="MathJax-Span-241" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-242" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub><mo>=</mo><mtext>min</mtext><mo stretchy="false">(</mo><msup><mi>T</mi><mo>+</mo></msup><mo>,</mo><msup><mi>T</mi><mo>−</mo></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-24">T_t = \text{min}(T^+, T^-)</script></span></li>
<li>Find the critical value, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-243" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-244"><span class="msubsup" id="MathJax-Span-245"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-246" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-247" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-25">T_c</script></span> from the table</li>
<li>Compare <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-248" style="width: 1.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.82em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-249"><span class="msubsup" id="MathJax-Span-250"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-251" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-252" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-26">T_t</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-253" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-254"><span class="msubsup" id="MathJax-Span-255"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-256" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-257" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-27">T_c</script></span>, rejecting <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-258" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-259"><span class="msubsup" id="MathJax-Span-260"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-261" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-262" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-28">H_0</script></span> if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-263" style="width: 1.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.82em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-264"><span class="msubsup" id="MathJax-Span-265"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-266" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-267" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>t</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-29">T_t</script></span> is smaller</li>
</ol><h3 id="mann-whitney-u-test"><a class="anchor hidden-xs" href="#mann-whitney-u-test" title="mann-whitney-u-test"><span class="octicon octicon-link"></span></a>Mann-Whitney U-test</h3><p>The Mann-Whitney U test tests whether two samples were taken from the same population. It is used when a t test cannot be applied as the data is not normal</p><ol>
<li>State the hypotheses<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-268" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-269"><span class="msubsup" id="MathJax-Span-270"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-271" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-272" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-30">H_0</script></span> The samples are from the same population, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B7;</mi><mn>1</mn></msub><mo>=</mo><msub><mi>&#x03B7;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-273" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1003.13em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-274"><span class="msubsup" id="MathJax-Span-275"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-276" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-277" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-278" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-279" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-280" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-281" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>η</mi><mn>1</mn></msub><mo>=</mo><msub><mi>η</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-31">\eta_1 = \eta_2</script></span><br>
ii. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-282" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-283"><span class="msubsup" id="MathJax-Span-284"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-285" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-286" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-32">H_1</script></span> The samples are from different populations, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B7;</mi><mn>1</mn></msub><mo>=</mo><msub><mi>&#x03B7;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-287" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1003.13em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-288"><span class="msubsup" id="MathJax-Span-289"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-290" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-291" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-292" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-293" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-294" style="font-family: STIXGeneral; font-style: italic;">η</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-295" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>η</mi><mn>1</mn></msub><mo>=</mo><msub><mi>η</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-33">\eta_1 = \eta_2</script></span></li>
<li>Rank the entire dataset and calculate the sum of the ranks for each set</li>
<li>Calculate the test statistic for each set <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mi>T</mi><mo>&#x2212;</mo><mfrac><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-296" style="width: 7.687em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.304em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1006.3em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-297"><span class="mi" id="MathJax-Span-298" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-299" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-300" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-301" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mfrac" id="MathJax-Span-302" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-303"><span class="mi" id="MathJax-Span-304" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-305" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-306" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-307" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-308" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-309" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-310" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.154em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mi>T</mi><mo>−</mo><mfrac><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-34">U = T - \frac{n(n+1)}{2}</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-311" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-312"><span class="mi" id="MathJax-Span-313" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-35">T</script></span> is the sum of the ranks of the set and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-314" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-315"><span class="mi" id="MathJax-Span-316" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-36">n</script></span> is the size of the set.<br> Let <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mtext>min</mtext><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-317" style="width: 8.558em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.021em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1006.97em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-318"><span class="mi" id="MathJax-Span-319" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-320" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mtext" id="MathJax-Span-321" style="font-family: STIXGeneral; padding-left: 0.31em;">min</span><span class="mo" id="MathJax-Span-322" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-323"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-324" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-325" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-326" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-327" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-328" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-329" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-330" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mtext>min</mtext><mo stretchy="false">(</mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><msub><mi>U</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-37">U = \text{min}(U_1, U_2)</script></span></li>
<li>Find the critical value for the significance level and the size of each data set</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-331" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.77em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-332"><span class="mi" id="MathJax-Span-333" style="font-family: STIXGeneral; font-style: italic;">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></span></span><script type="math/tex" id="MathJax-Element-38">U</script></span> is less than the critical value, reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-334" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-335"><span class="msubsup" id="MathJax-Span-336"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-337" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-338" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-39">H_0</script></span></li>
</ol><h3 id="kruskal-wallis-test"><a class="anchor hidden-xs" href="#kruskal-wallis-test" title="kruskal-wallis-test"><span class="octicon octicon-link"></span></a>Kruskal-Wallis test</h3><p>The Kruskal-Wallis test is a non-parametric version of the ANOVA test.<br>
It determines a difference between samples</p><ol>
<li>State the hypotheses<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-339" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-340"><span class="msubsup" id="MathJax-Span-341"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-342" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-343" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-40">H_0</script></span> All samples are from the same population<br>
ii. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-344" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-345"><span class="msubsup" id="MathJax-Span-346"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-347" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-348" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-41">H_1</script></span> Samples are from different populations</li>
<li>Rank the entire dataset and calculate the sum of the ranks of each set</li>
<li>Calculate the test statistic <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mn>12</mn><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>&#x2211;</mo><mfrac><msubsup><mi>T</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>i</mi></msub></mfrac></mrow><mo>)</mo></mrow><mo>&#x2212;</mo><mn>3</mn><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-349" style="width: 16.089em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.169em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.205em 1013.12em 4.357em -999.997em); top: -3.532em; left: 0em;"><span class="mrow" id="MathJax-Span-350"><span class="mi" id="MathJax-Span-351" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-352" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-353" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-354" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">(</span></span><span class="mrow" id="MathJax-Span-355"><span class="mfrac" id="MathJax-Span-356"><span style="display: inline-block; position: relative; width: 2.462em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.67em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.356em;"><span class="mn" id="MathJax-Span-357" style="font-size: 70.7%; font-family: STIXGeneral;">12</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1002.31em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -1.176em;"><span class="mrow" id="MathJax-Span-358"><span class="mi" id="MathJax-Span-359" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-360" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-361" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-362" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-363" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-364" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.462em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-365" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.207em;">∑</span><span class="mfrac" id="MathJax-Span-366" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.23em 1000.82em 4.357em -999.997em); top: -4.608em; left: 50%; margin-left: -0.407em;"><span class="msubsup" id="MathJax-Span-367"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-368" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.515em;"><span class="mn" id="MathJax-Span-369" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.21em 4.152em -999.997em); top: -3.788em; left: 0.412em;"><span class="mi" id="MathJax-Span-370" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.57em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.254em;"><span class="msubsup" id="MathJax-Span-371"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-372" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-373" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.92em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.925em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-374" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">)</span></span></span><span class="mo" id="MathJax-Span-375" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-376" style="font-family: STIXGeneral; padding-left: 0.259em;">3</span><span class="mo" id="MathJax-Span-377" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-378" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-379" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-380" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-381" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.537em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.441em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mn>12</mn><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>∑</mo><mfrac><msubsup><mi>T</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>i</mi></msub></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-42">H = \left(\frac{12}{N(N+1)}\sum \frac{T_i^2}{n_i}\right) - 3(N+1)</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-382" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-383"><span class="mi" id="MathJax-Span-384" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math></span></span><script type="math/tex" id="MathJax-Element-43">N</script></span> is the sum of all samples sizes, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-385" style="width: 1.027em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.82em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-386"><span class="msubsup" id="MathJax-Span-387"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-388" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mi" id="MathJax-Span-389" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-44">T_i</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-390" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.77em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-391"><span class="msubsup" id="MathJax-Span-392"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-393" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-394" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-45">n_i</script></span> are the rank sums and sizes of each sample</li>
<li>Find the degrees of freedom, the number of samples minus one</li>
<li>Calculate the critical value for the given significance level either from the tables or with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CHISQUARE_ICDF</mtext><mo stretchy="false">(</mo><mi>&#x03BD;</mi><mo>,</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-395" style="width: 15.013em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.298em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1012.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-396"><span class="mtext" id="MathJax-Span-397" style="font-family: STIXGeneral;">CHISQUARE_ICDF</span><span class="mo" id="MathJax-Span-398" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-399" style="font-family: STIXGeneral; font-style: italic;">ν</span><span class="mo" id="MathJax-Span-400" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-401" style="font-family: STIXGeneral; padding-left: 0.207em;">1</span><span class="mo" id="MathJax-Span-402" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-403" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-404" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CHISQUARE_ICDF</mtext><mo stretchy="false">(</mo><mi>ν</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-46">\text{CHISQUARE_ICDF}(\nu, 1-\alpha)</script></span></li>
<li>If the test statistic is larger than the critical value, reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-405" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-406"><span class="msubsup" id="MathJax-Span-407"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-408" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-409" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-47">H_0</script></span></li>
</ol><h2 id="correlation"><a class="anchor hidden-xs" href="#correlation" title="correlation"><span class="octicon octicon-link"></span></a>Correlation</h2><h3 id="spearman’s-rank-correlation-coefficient"><a class="anchor hidden-xs" href="#spearman’s-rank-correlation-coefficient" title="spearman’s-rank-correlation-coefficient"><span class="octicon octicon-link"></span></a>Spearman’s Rank Correlation Coefficient</h3><p>Spearman’s rank correlation coefficient is used when the data is ranked.</p><p>The value is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn><mo>&#x2212;</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>6</mn><mo>&#x2211;</mo><msup><mi>d</mi><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-410" style="width: 9.582em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.841em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1007.84em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-411"><span class="msubsup" id="MathJax-Span-412"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-413" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-414" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-415" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-416" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span><span class="mo" id="MathJax-Span-417" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mstyle" id="MathJax-Span-418" style="padding-left: 0.259em;"><span class="mrow" id="MathJax-Span-419"><span class="mfrac" id="MathJax-Span-420"><span style="display: inline-block; position: relative; width: 3.896em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1002.82em 4.408em -999.997em); top: -4.711em; left: 50%; margin-left: -1.432em;"><span class="mrow" id="MathJax-Span-421"><span class="mn" id="MathJax-Span-422" style="font-family: STIXGeneral;">6</span><span class="mo" id="MathJax-Span-423" style="font-family: STIXGeneral; vertical-align: 0.003em; padding-left: 0.207em;">∑</span><span class="msubsup" id="MathJax-Span-424" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-425" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-426" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1003.74em 4.306em -999.997em); top: -3.276em; left: 50%; margin-left: -1.893em;"><span class="mrow" id="MathJax-Span-427"><span class="mi" id="MathJax-Span-428" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-429" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-430"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-431" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.515em;"><span class="mn" id="MathJax-Span-432" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-433" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-434" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-435" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.9em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.896em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>s</mi></msub><mo>=</mo><mn>1</mn><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mn>6</mn><mo>∑</mo><msup><mi>d</mi><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></span></span><script type="math/tex" id="MathJax-Element-48">r_s = 1- \dfrac{6\sum d^2}{n(n^2-1)}</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-436" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.77em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-437"><span class="msubsup" id="MathJax-Span-438"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-439" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-440" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-49">d_i</script></span> is the difference between the rank of a pair of values, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-441" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-442"><span class="mi" id="MathJax-Span-443" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-50">n</script></span> is the (equal) size of each dataset</p><h4 id="testing-with-the-correlation-coefficient"><a class="anchor hidden-xs" href="#testing-with-the-correlation-coefficient" title="testing-with-the-correlation-coefficient"><span class="octicon octicon-link"></span></a>Testing with the correlation coefficient</h4><p>A test can be carried out in order to determine whether there is or is not a correlation between two random variables.<br>
Assuming that the correlation coefficient, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>s</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-444" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.72em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-445"><span class="msubsup" id="MathJax-Span-446"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-447" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-448" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mi>s</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-51">r_s</script></span>, is already known.</p><ol>
<li>State the hypotheses<br>
i. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mspace width="1em" /><mi>&#x03C1;</mi><mo>=</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-449" style="width: 6.662em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.433em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.43em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-450"><span class="msubsup" id="MathJax-Span-451"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-452" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-453" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-454" style="font-family: STIXGeneral; padding-left: 0.31em;">:</span><span class="mspace" id="MathJax-Span-455" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-456" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ρ</span><span class="mo" id="MathJax-Span-457" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-458" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mspace width="1em"></mspace><mi>ρ</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-52">H_0:\quad \rho = 0</script></span> (independent)<br>
ii. <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo>:</mo><mspace width="1em" /><mi>&#x03C1;</mi><mo>&#x2260;</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-459" style="width: 6.662em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.433em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.43em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-460"><span class="msubsup" id="MathJax-Span-461"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-462" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-463" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-464" style="font-family: STIXGeneral; padding-left: 0.31em;">:</span><span class="mspace" id="MathJax-Span-465" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-466" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ρ</span><span class="mo" id="MathJax-Span-467" style="font-family: STIXGeneral; padding-left: 0.31em;">≠</span><span class="mn" id="MathJax-Span-468" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo>:</mo><mspace width="1em"></mspace><mi>ρ</mi><mo>≠</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-53">H_1:\quad \rho \neq 0</script></span> (not-independent)</li>
<li>Find the critical value from the table</li>
<li>If the correlation coefficient is greater than the critical value, reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-469" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-470"><span class="msubsup" id="MathJax-Span-471"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-472" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-473" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-54">H_0</script></span></li>
</ol><h1 id="ss04"><a class="anchor hidden-xs" href="#ss04" title="ss04"><span class="octicon octicon-link"></span></a>SS04</h1><h2 id="linear-combinations-of-independent-normal-variables"><a class="anchor hidden-xs" href="#linear-combinations-of-independent-normal-variables" title="linear-combinations-of-independent-normal-variables"><span class="octicon octicon-link"></span></a>Linear combinations of independent normal variables</h2><p>If the letters <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-474" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-475"><span class="mi" id="MathJax-Span-476" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-55">X</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-477" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-478"><span class="mi" id="MathJax-Span-479" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math></span></span><script type="math/tex" id="MathJax-Element-56">Y</script></span> are variables and the letters <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-480" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-481"><span class="mi" id="MathJax-Span-482" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-57">a</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-483" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-484"><span class="mi" id="MathJax-Span-485" style="font-family: STIXGeneral; font-style: italic;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-58">b</script></span> are constants then</p><ol>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-486" style="width: 11.683em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.582em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1009.53em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-487"><span class="mi" id="MathJax-Span-488" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-489" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-490" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-491" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-492" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-493" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-494" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-495" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-496" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">a</span><span class="mi" id="MathJax-Span-497" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-498" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-499" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-500" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-501" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-502" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-59">E(aX + b) = aE(X) + b</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-503" style="width: 12.451em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.197em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1010.15em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-504"><span class="mi" id="MathJax-Span-505" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-506" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-507" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-508" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-509" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-510" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-511" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-512" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-513" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-514" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-515" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-516" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-517" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-518" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-519" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-520" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-60">E(X + Y) = E(X) + E(Y)</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-521" style="width: 36.888em; display: inline-block;"><span style="display: inline-block; position: relative; width: 30.228em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1030.18em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-522"><span class="mi" id="MathJax-Span-523" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-524" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-525"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-526" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-527" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-528" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-529" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-530" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-531" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-532"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-533" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-534" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-535" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-536" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-537" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-538" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-539"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-540" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-541" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-542" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-543" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-544" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-545" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-546" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-547"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-548" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-549" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-550"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-551" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-552" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-553" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-554" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-555" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-556" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-557" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-558" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-559" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-560" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-561" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-562" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-563" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-564"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-565" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-566" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-567" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-568" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-569" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-570" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-571" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-572" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-573" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-574"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-575" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-576" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-577" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-578" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-579" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-580" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-581" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-582" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-583"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-584" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-585" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-586" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-587" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-588"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-589" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-590" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-591" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-61">E(a_0 + a_1X_1 + a_2X_2 + ... + a_nX_n) = a_0 + a_1E(X_1) + a_2E(X_2) + ... + a_nE(X_n)</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-592" style="width: 12.605em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.3em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1010.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-593"><span class="mi" id="MathJax-Span-594" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-595" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-596" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-597" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-598" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-599" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-600" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mi" id="MathJax-Span-601" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-602" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-603" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-604" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-605" style="font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-606" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-607" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-608" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-609" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-610" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-611" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-612" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-62">Var(a+bX) = b^2Var(X)</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&#x00B1;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-613" style="width: 16.089em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.169em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1013.12em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-614"><span class="mi" id="MathJax-Span-615" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-616" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-617" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-618" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-619" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-620" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mi" id="MathJax-Span-621" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-622" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-623" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-624" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-625" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-626" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-627" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-628" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-629" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-630" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-631" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-632" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-633" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-634" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-635" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-636" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo>±</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-63">Var(X\pm Y) = Var(X) + Var(Y)</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msubsup><mi>a</mi><mi>n</mi><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-637" style="width: 39.142em; display: inline-block;"><span style="display: inline-block; position: relative; width: 32.072em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1032.02em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-638"><span class="mi" id="MathJax-Span-639" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-640" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-641" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-642" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-643"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-644" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-645" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-646" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-647" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-648" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-649" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-650"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-651" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-652" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-653" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-654" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-655" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-656" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-657"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-658" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-659" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-660" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-661" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-662" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-663" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-664" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-665"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-666" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-667" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-668"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-669" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-670" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-671" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-672" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-673" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-674" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-675" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mn" id="MathJax-Span-676" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-677" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-678" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-679" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-680" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-681"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-682" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-683" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-684" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-685" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-686" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-687" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-688" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mn" id="MathJax-Span-689" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-690" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-691" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-692" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-693" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-694"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-695" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-696" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-697" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-698" style="font-family: STIXGeneral;">+</span><span class="mo" id="MathJax-Span-699" style="font-family: STIXGeneral;">.</span><span class="mo" id="MathJax-Span-700" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-701" style="font-family: STIXGeneral; padding-left: 0.207em;">.</span><span class="mo" id="MathJax-Span-702" style="font-family: STIXGeneral; padding-left: 0.207em;">+</span><span class="msubsup" id="MathJax-Span-703"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-704" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-705" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-706" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-707" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-708" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-709" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-710" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-711"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-712" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-713" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-714" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>a</mi><mn>1</mn><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>a</mi><mn>2</mn><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msubsup><mi>a</mi><mi>n</mi><mn>2</mn></msubsup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-64">Var(a_0 + a_1X_1 + a_2X_2 + ... + a_nX_n) = a_1^2Var(X_1) + a_2^2Var(X_2) + ... + a_n^2Var(X_n)</script></span></li>
<li>A linear combination of independent, normal variables will itself be normally distributed</li>
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top: -2.866em; left: 0.156em;"><span class="texatom" id="MathJax-Span-722"><span class="mrow" id="MathJax-Span-723"><span class="mi" id="MathJax-Span-724" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span class="mo" id="MathJax-Span-725" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-726" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-727" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-728" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-729" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-730" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-731" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-732" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-733" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-734"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-735" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-736" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-737" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-738" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-739" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-740" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-741" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-742" style="font-family: STIXGeneral;">)</span></span><span class="mo" id="MathJax-Span-743" style="vertical-align: -0.766em;"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span class="mo" id="MathJax-Span-744" style="font-family: STIXGeneral; 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top: -2.866em; left: 0.156em;"><span class="texatom" id="MathJax-Span-751"><span class="mrow" id="MathJax-Span-752"><span class="mi" id="MathJax-Span-753" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span class="mo" id="MathJax-Span-754" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-755" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-756" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-757" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-758" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-759" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-760"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-761" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; 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font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.464em;"><span class="mn" id="MathJax-Span-774" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -3.686em; left: 0.464em;"><span class="mi" id="MathJax-Span-775" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-776"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-777" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-778" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-779" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-780" style="vertical-align: -0.766em;"><span style="font-family: STIXSizeFourSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>c</mi><mi>i</mi></msub><mo>×</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>i</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mo>∼</mo><mi>N</mi><mrow><mo>(</mo><mrow><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>c</mi><mi>i</mi></msub><msub><mi>μ</mi><mi>i</mi></msub><mo>,</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>c</mi><mi>i</mi><mn>2</mn></msubsup><msubsup><mi>σ</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-65">\left(\sum_{i=1}^n c_i\times N(\mu_i, \sigma_i^2) \right)\sim N \left(\sum_{i=1}^n c_i\mu_i, \sum_{i=1}^n c_i^2\sigma_i^2\right)</script></span></li>
</ol><p>Given a normal distribution <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mo>,</mo><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-781" style="width: 6.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.33em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1005.28em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-782"><span class="mi" id="MathJax-Span-783" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-784" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-785" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-786" style="font-family: STIXGeneral;">(</span><span class="munderover" id="MathJax-Span-787"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-788" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-789" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-790" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-791" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-792" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-793" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-794" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><mo>,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-66">X\sim N(\overline{x}, \sigma^2)</script></span>, if an event is given which is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-67-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-795" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-796"><span class="mi" id="MathJax-Span-797" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-67">k</script></span> times <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-68-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-798" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-799"><span class="mi" id="MathJax-Span-800" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-68">X</script></span> then this new event has a distribution <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-69-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mi>k</mi><mo>&#x00D7;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mo>,</mo><mi>k</mi><mo>&#x00D7;</mo><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-801" style="width: 7.994em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1006.51em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-802"><span class="mi" id="MathJax-Span-803" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-804" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-805" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-806" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="munderover" id="MathJax-Span-807" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-808" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-809" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-810" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-811" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-812" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-813" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-814" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-815" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-816" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mi>k</mi><mo>×</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><mo>,</mo><mi>k</mi><mo>×</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-69">N(k\times \overline{x}, k \times \sigma^2)</script></span>. The standard deviation of the new event is then <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-70-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>k</mi><mo>&#x00D7;</mo><msup><mi>&#x03C3;</mi><mn>2</mn></msup></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-817" style="width: 4.152em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.384em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1003.38em 2.462em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-818"><span class="msqrt" id="MathJax-Span-819"><span style="display: inline-block; position: relative; width: 3.384em; height: 0px;"><span style="position: absolute; clip: rect(3.076em 1002.67em 4.152em -999.997em); top: -3.993em; left: 0.72em;"><span class="mrow" id="MathJax-Span-820"><span class="mi" id="MathJax-Span-821" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-822" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-823" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-824" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.617em;"><span class="mn" id="MathJax-Span-825" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.67em 3.384em -999.997em); top: -3.993em; left: 0.72em;"><span style="display: inline-block; position: relative; width: 2.666em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 2.154em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.873em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.283em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.744em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.923em 1000.77em 4.152em -999.997em); top: -3.891em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>k</mi><mo>×</mo><msup><mi>σ</mi><mn>2</mn></msup></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-70">\sqrt{k \times \sigma^2}</script></span>.</p><p>Given two normal distributions <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-71-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mi>A</mi></msub><mo>,</mo><msubsup><mi>&#x03C3;</mi><mi>A</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-826" style="width: 7.072em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.791em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1005.74em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-827"><span class="mi" id="MathJax-Span-828" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mo" id="MathJax-Span-829" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-830" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-831" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-832"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="munderover" id="MathJax-Span-833"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-834" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-835" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-836" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-837" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-838" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-839" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-840" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-841" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-842" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mover><mi>x</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>A</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-71">A\sim N(\overline{x}_A, \sigma^2_A)</script></span>, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-72-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mi>B</mi></msub><mo>,</mo><msubsup><mi>&#x03C3;</mi><mi>B</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-843" style="width: 7.072em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.791em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1005.74em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-844"><span class="mi" id="MathJax-Span-845" style="font-family: STIXGeneral; font-style: italic;">B</span><span class="mo" id="MathJax-Span-846" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-847" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-848" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-849"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="munderover" id="MathJax-Span-850"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-851" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-852" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-853" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-854" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-855" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-856" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-857" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-858" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-859" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mover><mi>x</mi><mo accent="false">¯</mo></mover><mi>B</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>B</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-72">B\sim N(\overline{x}_B, \sigma^2_B)</script></span> the normally distributed random variable <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-73-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>&#x00B1;</mo><mi>B</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-860" style="width: 2.974em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.41em 2.82em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-861"><span class="mi" id="MathJax-Span-862" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mo" id="MathJax-Span-863" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mi" id="MathJax-Span-864" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">B</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>±</mo><mi>B</mi></math></span></span><script type="math/tex" id="MathJax-Element-73">A \pm B</script></span> has the distribution <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-74-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow><mo>(</mo><mrow><msub><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mi>A</mi></msub><mo>&#x00B1;</mo><msub><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mi>B</mi></msub><mo>,</mo><msubsup><mi>&#x03C3;</mi><mi>A</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>&#x03C3;</mi><mi>B</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-865" style="width: 10.761em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.814em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1008.66em 3.486em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-866"><span class="mi" id="MathJax-Span-867" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mrow" id="MathJax-Span-868" style="padding-left: 0.207em;"><span class="mo" id="MathJax-Span-869" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">(</span></span><span class="mrow" id="MathJax-Span-870"><span class="msubsup" id="MathJax-Span-871"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="munderover" id="MathJax-Span-872"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-873" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-874" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-875" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-876" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="msubsup" id="MathJax-Span-877" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="munderover" id="MathJax-Span-878"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-879" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-880" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-881" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-882" style="font-family: STIXGeneral;">,</span><span class="msubsup" id="MathJax-Span-883" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-884" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-885" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-886" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-887" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-888" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-889" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-890" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-891" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-892" style="vertical-align: -0.202em;"><span style="font-family: STIXSizeOneSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.628em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mrow><mo>(</mo><mrow><msub><mover><mi>x</mi><mo accent="false">¯</mo></mover><mi>A</mi></msub><mo>±</mo><msub><mover><mi>x</mi><mo accent="false">¯</mo></mover><mi>B</mi></msub><mo>,</mo><msubsup><mi>σ</mi><mi>A</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mi>B</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-74">N\left(\overline{x}_A \pm \overline{x}_B, \sigma^2_A + \sigma^2_B\right)</script></span>.</p><h2 id="approximating-distributions"><a class="anchor hidden-xs" href="#approximating-distributions" title="approximating-distributions"><span class="octicon octicon-link"></span></a>Approximating distributions</h2><ol>
<li>The purpose of making an approximation is:
<ul>
<li>To reduce the amount of calculation</li>
<li>To allow tables to be used where they otherwise could not</li>
<li>To calculate confidence intervals</li>
</ul>
</li>
<li>The binomial distribution may be approximated by the Poisson distribution if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-75-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2265;</mo><mn>50</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-893" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.82em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-894"><span class="mi" id="MathJax-Span-895" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-896" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mn" id="MathJax-Span-897" style="font-family: STIXGeneral; padding-left: 0.31em;">50</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>50</mn></math></span></span><script type="math/tex" id="MathJax-Element-75">n \geq 50</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-76-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#x2264;</mo><mn>0.1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-898" style="width: 3.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.076em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.97em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-899"><span class="mi" id="MathJax-Span-900" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-901" style="font-family: STIXGeneral; padding-left: 0.31em;">≤</span><span class="mn" id="MathJax-Span-902" style="font-family: STIXGeneral; padding-left: 0.31em;">0.1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≤</mo><mn>0.1</mn></math></span></span><script type="math/tex" id="MathJax-Element-76">p \leq 0.1</script></span></li>
<li>The conditions for the approximations are rules of theu,b. They are not shar dividing lines between good approximations and bad approximations</li>
<li>The binomial distribution may be approximated by the normal distribution if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-77-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2265;</mo><mn>50</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-903" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.82em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-904"><span class="mi" id="MathJax-Span-905" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-906" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mn" id="MathJax-Span-907" style="font-family: STIXGeneral; padding-left: 0.31em;">50</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>50</mn></math></span></span><script type="math/tex" id="MathJax-Element-77">n \geq 50</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-78-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>p</mi><mo>&#x2265;</mo><mn>10</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-908" style="width: 3.998em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.28em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-909"><span class="mi" id="MathJax-Span-910" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-911" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-912" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mn" id="MathJax-Span-913" style="font-family: STIXGeneral; padding-left: 0.31em;">10</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>p</mi><mo>≥</mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-78">np \geq 10</script></span></li>
<li>The Poisson distribution may be approximated by the normal distribution if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-79-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi><mo>&#x2265;</mo><mn>10</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-914" style="width: 3.384em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.77em 2.82em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-915"><span class="mi" id="MathJax-Span-916" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-917" style="font-family: STIXGeneral; padding-left: 0.31em;">≥</span><span class="mn" id="MathJax-Span-918" style="font-family: STIXGeneral; padding-left: 0.31em;">10</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.184em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>≥</mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-79">\lambda \geq 10</script></span></li>
</ol><h2 id="confidence-intervals"><a class="anchor hidden-xs" href="#confidence-intervals" title="confidence-intervals"><span class="octicon octicon-link"></span></a>Confidence intervals</h2><ol>
<li>An estimate of a population standard deviation calculated from a random sample of size <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-80-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-919" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-920"><span class="mi" id="MathJax-Span-921" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-80">n</script></span> has <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-81-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-922" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-923"><span class="mi" id="MathJax-Span-924" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-925" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-926" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-81">n-1</script></span> degrees of freedom</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-82-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo accent="false">&#x00AF;</mo></mover></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-927" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1000.72em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-928"><span class="munderover" id="MathJax-Span-929"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-930" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-931" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-82">\overline{X}</script></span> is the mean of a random sample of size <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-83-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-932" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-933"><span class="mi" id="MathJax-Span-934" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-83">n</script></span> from a normal distribution with mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-84-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-935" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-936"><span class="mi" id="MathJax-Span-937" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-84">\mu</script></span> a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-85-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-938" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1005.13em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-939"><span class="mn" id="MathJax-Span-940" style="font-family: STIXGeneral;">100</span><span class="mo" id="MathJax-Span-941" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-942" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-943" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-944" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-945" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-946" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-85">100(1-\alpha)\%</script></span> confidence interval for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-86-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-947" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-948"><span class="mi" id="MathJax-Span-949" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-86">\mu</script></span> is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-87-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x00B1;</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mn>2</mn></mfrac></mrow><mo>,</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-950" style="width: 6.457em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.279em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1005.28em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-951"><span class="munderover" id="MathJax-Span-952"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-953" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.46em 3.998em -999.997em); top: -4.506em; left: 0.054em;"><span class="mo" id="MathJax-Span-954" style=""><span style="display: inline-block; position: relative; width: 0.464em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.259em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-955" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="msubsup" id="MathJax-Span-956" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.359em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-957" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.788em; left: 0.259em;"><span class="texatom" id="MathJax-Span-958"><span class="mrow" id="MathJax-Span-959"><span class="texatom" id="MathJax-Span-960"><span class="mrow" id="MathJax-Span-961"><span class="mfrac" id="MathJax-Span-962"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-963" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">α</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-964" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.41em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.412em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-965" style="font-size: 70.7%; font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-966" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-967" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-968" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-969"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-970" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.357em -999.997em); top: -3.584em; left: 50%; margin-left: -0.459em;"><span class="msqrt" id="MathJax-Span-971"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-972"><span class="mi" id="MathJax-Span-973" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.36em 1.283em -999.997em); top: -1.586em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.788em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.98em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.976em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 1.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo accent="false">¯</mo></mover><mo>±</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-87">\overline{x} \pm t_{{\frac \alpha 2}, n-1} \frac{s}{\sqrt{n}}</script></span></li>
</ol><h2 id="further-confidence-intervals"><a class="anchor hidden-xs" href="#further-confidence-intervals" title="further-confidence-intervals"><span class="octicon octicon-link"></span></a>Further confidence intervals</h2><ol>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-88-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-974" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-975"><span class="mi" id="MathJax-Span-976" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-88">x</script></span> is an observation from a Poisson distribution with mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-89-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-977" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-978"><span class="mi" id="MathJax-Span-979" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-89">\lambda</script></span> then an approximate <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-90-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-980" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1005.13em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-981"><span class="mn" id="MathJax-Span-982" style="font-family: STIXGeneral;">100</span><span class="mo" id="MathJax-Span-983" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-984" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-985" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-986" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-987" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-988" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-90">100(1-\alpha)\%</script></span> confidence interval for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-91-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-989" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-990"><span class="mi" id="MathJax-Span-991" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-91">\lambda</script></span> is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-92-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&#x00B1;</mo><msub><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mn>2</mn></mfrac></mrow></msub><msqrt><mi>x</mi></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-992" style="width: 4.767em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.896em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1003.9em 3.23em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-993"><span class="mi" id="MathJax-Span-994" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-995" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="msubsup" id="MathJax-Span-996" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.36em 4.255em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-997" style="font-family: STIXGeneral; font-style: italic;">z</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.788em; left: 0.412em;"><span class="texatom" id="MathJax-Span-998"><span class="mrow" id="MathJax-Span-999"><span class="mfrac" id="MathJax-Span-1000"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1001" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">α</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1002" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.41em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.412em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msqrt" id="MathJax-Span-1003"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.72em;"><span class="mrow" id="MathJax-Span-1004"><span class="mi" id="MathJax-Span-1005" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.46em 1.283em -999.997em); top: -1.842em; left: 0.72em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(2.923em 1000.77em 4.152em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.684em; border-left: 0px solid; width: 0px; height: 1.628em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>±</mo><msub><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow></msub><msqrt><mi>x</mi></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-92">x \pm z_{\frac \alpha 2}\sqrt{x}</script></span>, provided that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-93-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1006" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1007"><span class="mi" id="MathJax-Span-1008" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-93">x</script></span> is reasonably large, say <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-94-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>20</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1009" style="width: 2.462em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1010"><span class="mo" id="MathJax-Span-1011" style="font-family: STIXGeneral;">></span><span class="mn" id="MathJax-Span-1012" style="font-family: STIXGeneral; padding-left: 0.31em;">20</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>20</mn></math></span></span><script type="math/tex" id="MathJax-Element-94">> 20</script></span></li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-95-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1013" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1014"><span class="mi" id="MathJax-Span-1015" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-95">r</script></span> is an observation from a binomial distribution with parameters <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-96-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1016" style="width: 1.744em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1001.39em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1017"><span class="mi" id="MathJax-Span-1018" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1019" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-1020" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-96">n, p</script></span> then an approximate <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-97-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1021" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1005.13em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1022"><span class="mn" id="MathJax-Span-1023" style="font-family: STIXGeneral;">100</span><span class="mo" id="MathJax-Span-1024" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1025" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1026" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1027" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-1028" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-1029" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-97">100(1-\alpha)\%</script></span> confidence interval for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-98-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1030" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1031"><span class="mi" id="MathJax-Span-1032" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-98">p</script></span> is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-99-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&#x00B1;</mo><msub><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mn>2</mn></mfrac></mrow></msub><msqrt><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1033" style="width: 7.636em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.253em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1006.25em 3.23em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1034"><span class="texatom" id="MathJax-Span-1035"><span class="mrow" id="MathJax-Span-1036"><span class="munderover" id="MathJax-Span-1037"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1038" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.105em;"><span style="height: 0em; vertical-align: 0em; width: 0.412em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1039" style="font-family: STIXGeneral;">̂ <span style="height: 0em; vertical-align: 0em; margin-left: -0.254em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1040" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="msubsup" id="MathJax-Span-1041" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.36em 4.255em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1042" style="font-family: STIXGeneral; font-style: italic;">z</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.788em; left: 0.412em;"><span class="texatom" id="MathJax-Span-1043"><span class="mrow" id="MathJax-Span-1044"><span class="mfrac" id="MathJax-Span-1045"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1046" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">α</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1047" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.41em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.412em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msqrt" id="MathJax-Span-1048"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px;"><span style="position: absolute; clip: rect(2.718em 1002.36em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-1049"><span class="mfrac" id="MathJax-Span-1050"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1002em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-1051"><span class="texatom" id="MathJax-Span-1052"><span class="mrow" id="MathJax-Span-1053"><span class="munderover" id="MathJax-Span-1054"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1055" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.096em; left: 0.054em;"><span style="height: 0em; vertical-align: 0em; width: 0.259em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1056" style="font-size: 70.7%; font-family: STIXGeneral;">̂ <span style="height: 0em; vertical-align: 0em; margin-left: -0.151em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1057" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1058" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1059" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="texatom" id="MathJax-Span-1060"><span class="mrow" id="MathJax-Span-1061"><span class="munderover" id="MathJax-Span-1062"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1063" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.096em; left: 0.054em;"><span style="height: 0em; vertical-align: 0em; width: 0.259em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1064" style="font-size: 70.7%; font-family: STIXGeneral;">̂ <span style="height: 0em; vertical-align: 0em; margin-left: -0.151em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1065" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1066" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.154em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.41em 3.384em -999.997em); top: -4.454em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.898em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.361em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.771em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.13em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.539em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.684em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mo>±</mo><msub><mi>z</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow></msub><msqrt><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-99">\hat{p} \pm z_{\frac \alpha 2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}</script></span>, provided <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-100-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1067" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1068"><span class="mi" id="MathJax-Span-1069" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-100">r</script></span> is reasonably large, say <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-101-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&gt;</mo><mn>20</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1070" style="width: 2.462em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1071"><span class="mo" id="MathJax-Span-1072" style="font-family: STIXGeneral;">></span><span class="mn" id="MathJax-Span-1073" style="font-family: STIXGeneral; padding-left: 0.31em;">20</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>20</mn></math></span></span><script type="math/tex" id="MathJax-Element-101">> 20</script></span></li>
</ol><h2 id="further-hypothesis-testing-for-means"><a class="anchor hidden-xs" href="#further-hypothesis-testing-for-means" title="further-hypothesis-testing-for-means"><span class="octicon octicon-link"></span></a>Further hypothesis testing for means</h2><p>To carry out a hypothesis test for a mean based on a sample from a normal distribution with an unknown standard deviation:</p><p>The test statistic is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-102-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x2212;</mo><mi>&#x03BC;</mi></mrow><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1074" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.925em 1001.54em 3.281em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1075"><span class="mfrac" id="MathJax-Span-1076"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1001.18em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-1077"><span class="munderover" id="MathJax-Span-1078"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1079" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.36em 4.05em -999.997em); top: -4.352em; left: 0.054em;"><span class="mo" id="MathJax-Span-1080" style=""><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.207em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1081" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1082" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.98em 4.613em -999.997em); top: -3.532em; left: 50%; margin-left: -0.51em;"><span class="mfrac" id="MathJax-Span-1083"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.21em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.1em;"><span class="mi" id="MathJax-Span-1084" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.62em 4.306em -999.997em); top: -3.686em; left: 50%; margin-left: -0.305em;"><span class="msqrt" id="MathJax-Span-1085"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -3.993em; left: 0.361em;"><span class="mrow" id="MathJax-Span-1086"><span class="mi" id="MathJax-Span-1087" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.26em 1.283em -999.997em); top: -1.432em; left: 0.361em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.3px solid; width: 0.259em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.84em; left: 0em;"><span><span style="font-size: 50%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.72em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.72em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mi>x</mi><mo accent="false">¯</mo></mover><mo>−</mo><mi>μ</mi></mrow><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-102">\frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-103-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msqrt><mfrac><mrow><mi mathvariant="normal">&#x03A3;</mi><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mover><mi>x</mi><mo accent="false">&#x00AF;</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1088" style="width: 6.765em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.535em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1005.53em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1089"><span class="mi" id="MathJax-Span-1090" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-1091" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msqrt" id="MathJax-Span-1092" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.845em; height: 0px;"><span style="position: absolute; clip: rect(2.666em 1002.77em 4.613em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-1093"><span class="mfrac" id="MathJax-Span-1094"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.23em 1002.41em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -1.176em;"><span class="mrow" id="MathJax-Span-1095"><span class="mi" id="MathJax-Span-1096" style="font-size: 70.7%; font-family: STIXGeneral;">Σ</span><span class="mo" id="MathJax-Span-1097" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1098" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1099" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="munderover" id="MathJax-Span-1100"><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1101" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.36em 4.05em -999.997em); top: -4.352em; left: 0.054em;"><span class="mo" id="MathJax-Span-1102" style=""><span style="display: inline-block; position: relative; width: 0.361em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.207em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.105em;"><span style="font-size: 50%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1103"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.21em 4.255em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1104" style="font-size: 70.7%; font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.259em;"><span class="mn" id="MathJax-Span-1105" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-1106"><span class="mi" id="MathJax-Span-1107" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1108" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1109" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.513em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.77em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 2.257em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.873em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.335em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.796em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msqrt><mfrac><mrow><mi mathvariant="normal">Σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mover><mi>x</mi><mo accent="false">¯</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-103">s = \sqrt{\frac{\Sigma (x- \overline{x})^2}{n-1}}</script></span></p><h2 id="hypothesis-tests-for-proportions-and-for-the-mean-of-a-poisson-distribution"><a class="anchor hidden-xs" href="#hypothesis-tests-for-proportions-and-for-the-mean-of-a-poisson-distribution" title="hypothesis-tests-for-proportions-and-for-the-mean-of-a-poisson-distribution"><span class="octicon octicon-link"></span></a>Hypothesis tests for proportions and for the mean of a Poisson distribution</h2><ol>
<li>To test hypotheses about a binomial population proportion, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-104-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1110" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1111"><span class="mi" id="MathJax-Span-1112" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-104">p</script></span>, either:<br>
a. Determine the cumulative binomial probability of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-105-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1113" style="width: 3.332em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.718em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.67em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1114"><span class="mi" id="MathJax-Span-1115" style="font-family: STIXGeneral; font-style: italic;">B</span><span class="mo" id="MathJax-Span-1116" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1117" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1118" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-1119" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">p</span><span class="mo" id="MathJax-Span-1120" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-105">B(n, p)</script></span><br>
State <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-106-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1121" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1122"><span class="msubsup" id="MathJax-Span-1123"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1124" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-1125" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-106">H_0</script></span> that the mean remains the same, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-107-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1126" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1127"><span class="msubsup" id="MathJax-Span-1128"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1129" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-1130" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-107">H_1</script></span> that it exhibits the expected change. Then find the probability that this value occurs, and reject <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-108-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1131" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1132"><span class="msubsup" id="MathJax-Span-1133"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1134" style="font-family: STIXGeneral; font-style: italic;">H<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-1135" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-108">H_0</script></span> if the probability is higher than the level of the test.<br>
b. use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-109-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&#x2212;</mo><mi>p</mi></mrow><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></mfrac></mstyle><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1136" style="width: 9.685em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.943em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.976em 1007.89em 4.511em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1137"><span class="mstyle" id="MathJax-Span-1138"><span class="mrow" id="MathJax-Span-1139"><span class="mfrac" id="MathJax-Span-1140"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.128em 1002.21em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-1141"><span class="texatom" id="MathJax-Span-1142"><span class="mrow" id="MathJax-Span-1143"><span class="munderover" id="MathJax-Span-1144"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1145" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.045em; left: 0.105em;"><span style="height: 0em; vertical-align: 0em; width: 0.412em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1146" style="font-family: STIXGeneral;">̂ <span style="height: 0em; vertical-align: 0em; margin-left: -0.254em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1147" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1148" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">p</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.564em 1003.49em 4.716em -999.997em); top: -2.764em; left: 50%; margin-left: -1.739em;"><span class="msqrt" id="MathJax-Span-1149"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px;"><span style="position: absolute; clip: rect(2.82em 1002.36em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-1150"><span class="mfrac" id="MathJax-Span-1151"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-1152"><span class="mi" id="MathJax-Span-1153" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-1154" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1155" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1156" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1157" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-1158" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1159" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.154em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.41em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.898em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.361em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.771em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.13em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.539em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.59em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.589em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1160" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-1161" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1162" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1163" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-1164" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-1165" style="font-family: STIXGeneral; padding-left: 0.207em;">1</span><span class="mo" id="MathJax-Span-1166" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.247em; border-left: 0px solid; width: 0px; height: 4.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>p</mi></mrow><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></mfrac></mstyle><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-109">\dfrac{\hat{p} -p}{\sqrt{\frac{p(1-p)}{n}}} \sim N(0, 1)</script></span><br>
The hypotheses are then dealt with like a regular normal hypothesis test.</li>
<li>To test hypotheses about a Poisson population mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-110-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1167" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1168"><span class="mi" id="MathJax-Span-1169" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-110">\lambda</script></span>, either<br>
a. Determine the cumulative Poisson probability of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-111-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>&#x03BB;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1170" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1171"><span class="msubsup" id="MathJax-Span-1172"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1173" style="font-family: STIXGeneral; font-style: italic;">P</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-1174" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1175" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1176" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1177" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-111">P_0(\lambda)</script></span>, or<br>
b. use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-112-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>&#x03BB;</mi><mo stretchy="false">&#x005E;</mo></mover></mrow><mo>&#x2212;</mo><mi>&#x03BB;</mi></mrow><msqrt><mi>&#x03BB;</mi></msqrt></mfrac><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mtext>&#xA0;</mtext><mi>&#x03BB;</mi><mo>&gt;</mo><mn>10</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1178" style="width: 11.017em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.019em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1009.02em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1179"><span class="mfrac" id="MathJax-Span-1180"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.128em 1001.13em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-1181"><span class="texatom" id="MathJax-Span-1182"><span class="mrow" id="MathJax-Span-1183"><span class="munderover" id="MathJax-Span-1184"><span style="display: inline-block; position: relative; width: 0.31em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1185" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.249em; left: 0.054em;"><span style="height: 0em; vertical-align: 0em; width: 0.259em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-1186" style="font-size: 70.7%; font-family: STIXGeneral;">̂ <span style="height: 0em; vertical-align: 0em; margin-left: -0.151em;"></span></span><span style="display: inline-block; overflow: hidden; height: 1px; width: 0em;"></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1187" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1188" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.82em 4.255em -999.997em); top: -3.532em; left: 50%; margin-left: -0.407em;"><span class="msqrt" id="MathJax-Span-1189"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-1190"><span class="mi" id="MathJax-Span-1191" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.31em 1.283em -999.997em); top: -1.688em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.31em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.891em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.23em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.232em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1192" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-1193" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1194" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1195" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-1196" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-1197" style="font-family: STIXGeneral; padding-left: 0.207em;">1</span><span class="mo" id="MathJax-Span-1198" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1199" style="font-family: STIXGeneral;">,</span><span class="mtext" id="MathJax-Span-1200" style="font-family: STIXGeneral; padding-left: 0.207em;"> </span><span class="mi" id="MathJax-Span-1201" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1202" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-1203" style="font-family: STIXGeneral; padding-left: 0.31em;">10</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 2.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mover><mi>λ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>λ</mi></mrow><msqrt><mi>λ</mi></msqrt></mfrac><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mi>λ</mi><mo>></mo><mn>10</mn></math></span></span><script type="math/tex" id="MathJax-Element-112">\frac{\hat{\lambda} - \lambda}{\sqrt{\lambda}} \sim N(0, 1),\ \lambda > 10</script></span></li>
</ol><h1 id="ss05"><a class="anchor hidden-xs" href="#ss05" title="ss05"><span class="octicon octicon-link"></span></a>SS05</h1><h2 id="continuous-probability-distributions"><a class="anchor hidden-xs" href="#continuous-probability-distributions" title="continuous-probability-distributions"><span class="octicon octicon-link"></span></a>Continuous probability distributions</h2><ol>
<li>The random variables <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-113-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1204" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1205"><span class="mi" id="MathJax-Span-1206" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-113">X</script></span> having probability density function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-114-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>&#x2212;</mo><mi>a</mi></mrow></mfrac><mo>,</mo></mtd><mtd><mi>a</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1207" style="width: 13.476em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.017em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.462em 1011.02em 5.33em -999.997em); top: -4.147em; left: 0em;"><span class="mrow" id="MathJax-Span-1208"><span class="mi" id="MathJax-Span-1209" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1210" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1211" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1212" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1213" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-1214" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-1215" style="vertical-align: -0.561em;"><span style="font-family: STIXSizeThreeSym;">{</span></span><span class="mtable" id="MathJax-Span-1216" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 6.97em; height: 0px;"><span style="position: absolute; clip: rect(2.308em 1001.74em 5.126em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(2.974em 1001.74em 4.664em -999.997em); top: -4.659em; left: 0em;"><span class="mtd" id="MathJax-Span-1217"><span class="mrow" id="MathJax-Span-1218"><span class="mfrac" id="MathJax-Span-1219"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1220" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.18em 4.255em -999.997em); top: -3.584em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-1221"><span class="mi" id="MathJax-Span-1222" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1223" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1224" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1225" style="font-family: STIXGeneral;">,</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.306em -999.997em); top: -3.122em; left: 0em;"><span class="mtd" id="MathJax-Span-1233"><span class="mrow" id="MathJax-Span-1234"><span class="mn" id="MathJax-Span-1235" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-1236" style="font-family: STIXGeneral;">,</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1004.05em 5.023em -999.997em); top: -3.993em; left: 2.923em;"><span style="display: inline-block; position: relative; width: 4.05em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1004.05em 4.152em -999.997em); top: -4.659em; left: 0em;"><span class="mtd" id="MathJax-Span-1226"><span class="mrow" id="MathJax-Span-1227"><span class="mi" id="MathJax-Span-1228" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1229" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-1230" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1231" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-1232" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">b</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1003.9em 4.152em -999.997em); top: -3.122em; left: 0em;"><span class="mtd" id="MathJax-Span-1237"><span class="mrow" id="MathJax-Span-1238"><span class="mtext" id="MathJax-Span-1239" style="font-family: STIXGeneral;">otherwise</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1240"></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.309em; border-left: 0px solid; width: 0px; height: 3.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo>,</mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo><</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-114">f(x) = \begin{cases}\frac{1}{b-a},& a < x < b \\ 0,& \text{otherwise} \end{cases}</script></span> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-115-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1241" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1242"><span class="mi" id="MathJax-Span-1243" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-115">a</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-116-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1244" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1245"><span class="mi" id="MathJax-Span-1246" style="font-family: STIXGeneral; font-style: italic;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-116">b</script></span> are constants, it is said to follow a rectangular distribution</li>
<li>The mean of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-117-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1247" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1248"><span class="mi" id="MathJax-Span-1249" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-117">X</script></span> is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-118-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1250" style="width: 4.306em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.537em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.49em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1251"><span class="mfrac" id="MathJax-Span-1252"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1253" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1254" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1255" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1256" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1257" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1258" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mo" id="MathJax-Span-1259" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-118">\frac 1 2 (a+b)</script></span> and the variance of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-119-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1260" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1261"><span class="mi" id="MathJax-Span-1262" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-119">X</script></span> is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-120-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>12</mn></mfrac><mo stretchy="false">(</mo><mi>b</mi><mo>&#x2212;</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1263" style="width: 5.33em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.357em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1004.36em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1264"><span class="mfrac" id="MathJax-Span-1265"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1266" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.67em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.356em;"><span class="mn" id="MathJax-Span-1267" style="font-size: 70.7%; font-family: STIXGeneral;">12</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1268" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1269" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1270" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1271" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span><span class="msubsup" id="MathJax-Span-1272"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1273" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-1274" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>12</mn></mfrac><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-120">\frac{1}{12}(b-a)^2</script></span></li>
<li>The exponential distribution has probability density function <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-121-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>&#x03BB;</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03BB;</mi><mi>x</mi></mrow></msup></mtd><mtd><mi>x</mi><mo>&gt;</mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1275" style="width: 13.527em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.068em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.257em 1011.07em 4.818em -999.997em); top: -3.788em; left: 0em;"><span class="mrow" id="MathJax-Span-1276"><span class="mi" id="MathJax-Span-1277" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1278" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1279" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1280" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1281" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-1282" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-1283" style="vertical-align: -0.459em;"><span><span style="font-size: 111%; font-family: STIXSizeTwoSym;">{</span></span></span><span class="mtable" id="MathJax-Span-1284" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 7.123em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1002.1em 4.869em -999.997em); top: -3.993em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.103em; height: 0px;"><span style="position: absolute; clip: rect(3.025em 1002.1em 4.152em -999.997em); top: -4.557em; left: 0em;"><span class="mtd" id="MathJax-Span-1285"><span class="mrow" id="MathJax-Span-1286"><span class="mi" id="MathJax-Span-1287" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="msubsup" id="MathJax-Span-1288"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1289" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1290"><span class="mrow" id="MathJax-Span-1291"><span class="mo" id="MathJax-Span-1292" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1293" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span class="mi" id="MathJax-Span-1294" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 0em;"><span class="mtd" id="MathJax-Span-1300"><span class="mrow" id="MathJax-Span-1301"><span class="mn" id="MathJax-Span-1302" style="font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.615em 1003.9em 4.869em -999.997em); top: -3.993em; left: 3.23em;"><span style="display: inline-block; position: relative; width: 3.896em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.21em 4.152em -999.997em); top: -4.557em; left: 0em;"><span class="mtd" id="MathJax-Span-1295"><span class="mrow" id="MathJax-Span-1296"><span class="mi" id="MathJax-Span-1297" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1298" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-1299" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1003.9em 4.152em -999.997em); top: -3.327em; left: 0em;"><span class="mtd" id="MathJax-Span-1303"><span class="mrow" id="MathJax-Span-1304"><span class="mtext" id="MathJax-Span-1305" style="font-family: STIXGeneral;">otherwise</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1306"></span></span></span><span style="display: inline-block; width: 0px; height: 3.794em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mi>λ</mi><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup></mtd><mtd><mi>x</mi><mo>></mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true"></mo></mrow></math></span></span><script type="math/tex" id="MathJax-Element-121">f(x) = \begin{cases}\lambda e^{-\lambda x} & x > 0 \\ 0 & \text{otherwise}\end{cases}</script></span></li>
<li>The exponential distribution with parameer <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-122-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1307" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1308"><span class="mi" id="MathJax-Span-1309" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-122">\lambda</script></span> has mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-123-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>&#x03BB;</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1310" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1000.67em 2.769em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1311"><span class="mfrac" id="MathJax-Span-1312"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1313" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1314" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.559em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>λ</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-123">\frac 1 \lambda</script></span> and standard deviation <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-124-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>&#x03BB;</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1315" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1000.67em 2.769em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1316"><span class="mfrac" id="MathJax-Span-1317"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1318" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1319" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.559em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>λ</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-124">\frac 1 \lambda</script></span></li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-125-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1320" style="width: 4.511em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.64em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1321"><span class="mi" id="MathJax-Span-1322" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-1323" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1324" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1325" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-1326" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1327" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo><</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-125">P(X < x)</script></span> is known as the cumulative distribution function and is usually denoted <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-126-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1328" style="width: 2.205em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.74em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1329"><span class="mi" id="MathJax-Span-1330" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1331" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1332" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1333" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-126">F(x)</script></span></li>
<li>For the exponential distribution with parameter <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-127-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1334" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1335"><span class="mi" id="MathJax-Span-1336" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-127">\lambda</script></span>, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-128-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>&#x2212;</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03BB;</mi><mi>x</mi></mrow></msup><mo>,</mo><mspace width="1em" /><mi>x</mi><mo>&gt;</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1337" style="width: 12.503em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.248em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1010.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1338"><span class="mi" id="MathJax-Span-1339" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1340" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1341" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1342" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1343" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1344" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span><span class="mo" id="MathJax-Span-1345" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-1346" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1347" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1348"><span class="mrow" id="MathJax-Span-1349"><span class="mo" id="MathJax-Span-1350" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1351" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span class="mi" id="MathJax-Span-1352" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1353" style="font-family: STIXGeneral;">,</span><span class="mspace" id="MathJax-Span-1354" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1355" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1356" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-1357" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo>,</mo><mspace width="1em"></mspace><mi>x</mi><mo>></mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-128">F(x) = 1-e^{-\lambda x},\quad x > 0</script></span></li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-129-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1358" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1359"><span class="mi" id="MathJax-Span-1360" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-129">a</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-130-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1361" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1362"><span class="mi" id="MathJax-Span-1363" style="font-family: STIXGeneral; font-style: italic;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-130">b</script></span> are two constants and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-131-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&lt;</mo><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1364" style="width: 2.82em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1365"><span class="mi" id="MathJax-Span-1366" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1367" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-1368" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo><</mo><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-131">a<b</script></span>, the probability that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-132-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1369" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1370"><span class="mi" id="MathJax-Span-1371" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-132">X</script></span> takes a value between <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-133-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1372" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1373"><span class="mi" id="MathJax-Span-1374" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-133">a</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-134-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1375" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1376"><span class="mi" id="MathJax-Span-1377" style="font-family: STIXGeneral; font-style: italic;">b</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math></span></span><script type="math/tex" id="MathJax-Element-134">b</script></span> is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-135-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1378" style="width: 5.945em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.869em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.82em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1379"><span class="mi" id="MathJax-Span-1380" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1381" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1382" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mo" id="MathJax-Span-1383" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1384" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1385" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1386" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1387" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1388" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-135">F(b) - F(a)</script></span></li>
<li>The intervals between successive events from a Poisson distribution with mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-136-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1389" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1390"><span class="mi" id="MathJax-Span-1391" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-136">\lambda</script></span> are distributed according to the exponential distribution with parameter <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-137-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1392" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1393"><span class="mi" id="MathJax-Span-1394" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-137">\lambda</script></span></li>
</ol><h2 id="estimation"><a class="anchor hidden-xs" href="#estimation" title="estimation"><span class="octicon octicon-link"></span></a>Estimation</h2><ol>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-138-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1395" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1000.98em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1396"><span class="msubsup" id="MathJax-Span-1397"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1398" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1399" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>S</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-138">S^2</script></span> denotes the variance estimate from a random sample of size <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-139-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1400" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1401"><span class="mi" id="MathJax-Span-1402" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-139">n</script></span> from a normal population with variance of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-140-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1403" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1001.03em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1404"><span class="msubsup" id="MathJax-Span-1405"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1406" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-1407" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-140">\sigma ^2</script></span>, then <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-141-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>&#x03C3;</mi><mn>2</mn></msup></mfrac><mo>&#x223C;</mo><msubsup><mi>&#x03C7;</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1408" style="width: 7.072em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.791em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1005.79em 3.179em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1409"><span class="mfrac" id="MathJax-Span-1410"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.23em 1002.36em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -1.176em;"><span class="mrow" id="MathJax-Span-1411"><span class="mo" id="MathJax-Span-1412" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1413" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1414" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1415" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1416" style="font-size: 70.7%; font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1417"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1418" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.249em; left: 0.412em;"><span class="mn" id="MathJax-Span-1419" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.72em 4.152em -999.997em); top: -3.532em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-1420"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1421" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.198em; left: 0.412em;"><span class="mn" id="MathJax-Span-1422" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.51em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.513em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1423" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1424" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.744em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1425" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1426" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.28em 4.255em -999.997em); top: -3.686em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1427"><span class="mrow" id="MathJax-Span-1428"><span class="mi" id="MathJax-Span-1429" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1430" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1431" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 2.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msubsup><mi>χ</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-141">\frac{(n-1)S^2}{\sigma ^2} \sim \chi^2_{n-1}</script></span></li>
<li>The <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-142-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C7;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1432" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1000.98em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1433"><span class="msubsup" id="MathJax-Span-1434"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1435" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1436" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-142">\chi^2</script></span> distribution is not symmetric so both lower and upper percentage points need to be read from tables</li>
<li>A <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-143-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo><mi mathvariant="normal">&#x0025;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1437" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1005.13em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1438"><span class="mn" id="MathJax-Span-1439" style="font-family: STIXGeneral;">100</span><span class="mo" id="MathJax-Span-1440" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1441" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1442" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1443" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-1444" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-1445" style="font-family: STIXGeneral;">%</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mi mathvariant="normal">%</mi></math></span></span><script type="math/tex" id="MathJax-Element-143">100(1-\alpha)\%</script></span> confidence interval for a normal population variance, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-144-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1446" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1001.03em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1447"><span class="msubsup" id="MathJax-Span-1448"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1449" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-1450" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-144">\sigma^2</script></span>, is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-145-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msubsup><mi>&#x03C7;</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>&#x2212;</mo><mfrac><mi>&#x03B1;</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup></mfrac></mstyle></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1451" style="width: 5.126em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.203em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.515em 1004.2em 3.845em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1452"><span class="mstyle" id="MathJax-Span-1453"><span class="mrow" id="MathJax-Span-1454"><span class="mfrac" id="MathJax-Span-1455"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1003.85em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.944em;"><span class="mrow" id="MathJax-Span-1456"><span class="mo" id="MathJax-Span-1457" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1458" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1459" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1460" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-1461" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1462"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1463" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1464" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002em 4.869em -999.997em); top: -3.225em; left: 50%; margin-left: -1.022em;"><span class="msubsup" id="MathJax-Span-1465"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1466" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1467" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1001.54em 4.46em -999.997em); top: -3.584em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1468"><span class="mrow" id="MathJax-Span-1469"><span class="mn" id="MathJax-Span-1470" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1471" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mfrac" id="MathJax-Span-1472"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1473" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">α</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1474" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.41em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.412em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.998em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.872em; border-left: 0px solid; width: 0px; height: 3.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msubsup><mi>χ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup></mfrac></mstyle></math></span></span><script type="math/tex" id="MathJax-Element-145">\dfrac{(n-1)S^2}{\chi^2_{1- \frac \alpha 2}}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-146-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msubsup><mi>&#x03C7;</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>&#x03B1;</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup></mfrac></mstyle></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1475" style="width: 5.126em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.203em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.515em 1004.2em 3.845em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1476"><span class="mstyle" id="MathJax-Span-1477"><span class="mrow" id="MathJax-Span-1478"><span class="mfrac" id="MathJax-Span-1479"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1003.85em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.944em;"><span class="mrow" id="MathJax-Span-1480"><span class="mo" id="MathJax-Span-1481" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1482" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1483" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1484" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-1485" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1486"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1487" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1488" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.18em 4.869em -999.997em); top: -3.225em; left: 50%; margin-left: -0.561em;"><span class="msubsup" id="MathJax-Span-1489"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1490" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1491" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.72em 4.46em -999.997em); top: -3.584em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1492"><span class="mrow" id="MathJax-Span-1493"><span class="mfrac" id="MathJax-Span-1494"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -4.352em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1495" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">α</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.26em 4.152em -999.997em); top: -3.686em; left: 50%; margin-left: -0.1em;"><span class="mn" id="MathJax-Span-1496" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.41em 1.232em -999.997em); top: -1.227em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.412em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.998em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.872em; border-left: 0px solid; width: 0px; height: 3.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msubsup><mi>χ</mi><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup></mfrac></mstyle></math></span></span><script type="math/tex" id="MathJax-Element-146">\dfrac{(n-1)S^2}{\chi^2_{\frac \alpha 2}}</script></span></li>
<li>Confidence limits for a normal population standard deviation, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-147-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1497" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1498"><span class="mi" id="MathJax-Span-1499" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-147">\sigma</script></span> are found by taking the square root of those calculated for the population variance</li>
</ol><h2 id="hypothesis-testing-one-sample-tests"><a class="anchor hidden-xs" href="#hypothesis-testing-one-sample-tests" title="hypothesis-testing-one-sample-tests"><span class="octicon octicon-link"></span></a>Hypothesis testing: one sample tests</h2><ol>
<li>To test hypotheses about a normal population variance, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-148-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1500" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1001.03em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1501"><span class="msubsup" id="MathJax-Span-1502"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1503" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-1504" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-148">\sigma^2</script></span> or standard deviation <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-149-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1505" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1506"><span class="mi" id="MathJax-Span-1507" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-149">\sigma</script></span>, use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-150-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>&#x03C3;</mi><mn>2</mn></msup></mfrac></mstyle><mo>&#x223C;</mo><msubsup><mi>&#x03C7;</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1508" style="width: 8.916em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.277em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.873em 1007.28em 3.486em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1509"><span class="mstyle" id="MathJax-Span-1510"><span class="mrow" id="MathJax-Span-1511"><span class="mfrac" id="MathJax-Span-1512"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1003.85em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.944em;"><span class="mrow" id="MathJax-Span-1513"><span class="mo" id="MathJax-Span-1514" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1515" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1516" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1517" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-1518" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1519"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1520" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1521" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1001.03em 4.152em -999.997em); top: -3.276em; left: 50%; margin-left: -0.51em;"><span class="msubsup" id="MathJax-Span-1522"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1523" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.617em;"><span class="mn" id="MathJax-Span-1524" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.998em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1525" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1526" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.744em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1527" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1528" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.28em 4.255em -999.997em); top: -3.686em; left: 0.464em;"><span class="texatom" id="MathJax-Span-1529"><span class="mrow" id="MathJax-Span-1530"><span class="mi" id="MathJax-Span-1531" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1532" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1533" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.997em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac></mstyle><mo>∼</mo><msubsup><mi>χ</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-150">\dfrac{(n-1)S^2}{\sigma^2}\sim \chi^2_{n-1}</script></span></li>
<li>To test hypotheses about a normal population with mean, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-151-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1534" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1535"><span class="mi" id="MathJax-Span-1536" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-151">\mu</script></span>, use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-152-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mover><mi>X</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x2212;</mo><mi>&#x03BC;</mi></mrow><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac></mfrac></mstyle><mo>&#x223C;</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1537" style="width: 6.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.638em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.72em 1005.64em 4.152em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1538"><span class="mstyle" id="MathJax-Span-1539"><span class="mrow" id="MathJax-Span-1540"><span class="mfrac" id="MathJax-Span-1541"><span style="display: inline-block; position: relative; width: 2.564em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.82em 1002.41em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.227em;"><span class="mrow" id="MathJax-Span-1542"><span class="munderover" id="MathJax-Span-1543"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1544" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-1545" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1546" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1547" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">μ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1001.23em 4.716em -999.997em); top: -3.122em; left: 50%; margin-left: -0.612em;"><span class="mfrac" id="MathJax-Span-1548"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1549" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.357em -999.997em); top: -3.584em; left: 50%; margin-left: -0.459em;"><span class="msqrt" id="MathJax-Span-1550"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-1551"><span class="mi" id="MathJax-Span-1552" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.36em 1.283em -999.997em); top: -1.586em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.788em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.98em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.976em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.56em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.564em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1553" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1554" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1555" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.259em;"><span class="texatom" id="MathJax-Span-1556"><span class="mrow" id="MathJax-Span-1557"><span class="mi" id="MathJax-Span-1558" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1559" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1560" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.809em; border-left: 0px solid; width: 0px; height: 3.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mover><mi>X</mi><mo accent="false">¯</mo></mover><mo>−</mo><mi>μ</mi></mrow><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac></mfrac></mstyle><mo>∼</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-152">\dfrac{\overline{X}- \mu}{\frac{S}{\sqrt{n}}} \sim t_{n-1}</script></span></li>
</ol><h2 id="hypothesis-testing-two-sample-tests"><a class="anchor hidden-xs" href="#hypothesis-testing-two-sample-tests" title="hypothesis-testing-two-sample-tests"><span class="octicon octicon-link"></span></a>Hypothesis testing: two-sample tests</h2><ol>
<li>To test hypotheses about the equality of two normal population variances, or standard deviations, use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-153-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>s</mi><mi>x</mi><mn>2</mn></msubsup><msubsup><mi>s</mi><mi>y</mi><mn>2</mn></msubsup></mfrac><mo>&#x223C;</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mi>x</mi></msub><mo>&#x2212;</mo><mn>1</mn><mo>,</mo><mtext>&#xA0;</mtext><msub><mi>n</mi><mi>y</mi></msub><mo>&#x2212;</mo><mn>1</mn></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1561" style="width: 7.636em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.253em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.335em 1006.25em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1562"><span class="mfrac" id="MathJax-Span-1563"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1000.57em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1564"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.26em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1565" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.259em;"><span class="mn" id="MathJax-Span-1566" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -3.891em; left: 0.259em;"><span class="mi" id="MathJax-Span-1567" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.57em 4.357em -999.997em); top: -3.481em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1568"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.26em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1569" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.259em;"><span class="mn" id="MathJax-Span-1570" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.26em 4.255em -999.997em); top: -3.891em; left: 0.259em;"><span class="mi" id="MathJax-Span-1571" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.72em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.72em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1572" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1573" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1574" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-1575"><span class="mrow" id="MathJax-Span-1576"><span class="msubsup" id="MathJax-Span-1577"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1578" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1579" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1580" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1581" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-1582" style="font-size: 70.7%; font-family: STIXGeneral;">,</span><span class="mtext" id="MathJax-Span-1583" style="font-size: 70.7%; font-family: STIXGeneral;"> </span><span class="msubsup" id="MathJax-Span-1584"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1585" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1586" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1587" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1588" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>s</mi><mi>x</mi><mn>2</mn></msubsup><msubsup><mi>s</mi><mi>y</mi><mn>2</mn></msubsup></mfrac><mo>∼</mo><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mi>x</mi></msub><mo>−</mo><mn>1</mn><mo>,</mo><mtext> </mtext><msub><mi>n</mi><mi>y</mi></msub><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-153">\frac{s^2_x}{s^2_y} \sim F_{n_x-1,\ n_y - 1}</script></span></li>
<li>To test hypotheses about the equality of (or given differece in) two normal population means, based upon independent random samples and known population variances use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-154-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mover><mi>X</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x2212;</mo><mover><mi>Y</mi><mo accent="false">&#x00AF;</mo></mover><mo stretchy="false">)</mo><mo>&#x2212;</mo><mo stretchy="false">(</mo><msub><mi>&#x03BC;</mi><mi>x</mi></msub><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><msqrt><mfrac><msubsup><mi>&#x03C3;</mi><mi>x</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>x</mi></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>&#x03C3;</mi><mi>y</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>y</mi></msub></mfrac></msqrt></mfrac></mstyle><mo>&#x223C;</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1589" style="width: 15.525em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.707em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.617em 1012.66em 5.126em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1590"><span class="mstyle" id="MathJax-Span-1591"><span class="mrow" id="MathJax-Span-1592"><span class="mfrac" id="MathJax-Span-1593"><span style="display: inline-block; position: relative; width: 8.353em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.82em 1008.2em 4.46em -999.997em); top: -4.762em; left: 50%; margin-left: -4.096em;"><span class="mrow" id="MathJax-Span-1594"><span class="mo" id="MathJax-Span-1595" style="font-family: STIXGeneral;">(</span><span class="munderover" id="MathJax-Span-1596"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1597" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-1598" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1599" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="munderover" id="MathJax-Span-1600" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1601" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-1602" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.412em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1603" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1604" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mo" id="MathJax-Span-1605" style="font-family: STIXGeneral; padding-left: 0.259em;">(</span><span class="msubsup" id="MathJax-Span-1606"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1607" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1608" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1609" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-1610" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1611" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1612" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1613" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1004.56em 5.074em -999.997em); top: -2.456em; left: 50%; margin-left: -2.252em;"><span class="msqrt" id="MathJax-Span-1614"><span style="display: inline-block; position: relative; width: 4.562em; height: 0px;"><span style="position: absolute; clip: rect(2.666em 1003.38em 4.716em -999.997em); top: -3.993em; left: 1.13em;"><span class="mrow" id="MathJax-Span-1615"><span class="mfrac" id="MathJax-Span-1616"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1000.72em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-1617"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1618" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.412em;"><span class="mn" id="MathJax-Span-1619" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.26em 4.152em -999.997em); top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1620" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.62em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1621"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1622" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1623" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.87em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.873em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1624" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mfrac" id="MathJax-Span-1625" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1000.72em 4.357em -999.997em); top: -4.608em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-1626"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1627" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.412em;"><span class="mn" id="MathJax-Span-1628" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.64em 1000.26em 4.255em -999.997em); top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1629" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.62em 4.357em -999.997em); top: -3.635em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1630"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1631" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1632" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.87em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.873em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1003.43em 3.384em -999.997em); top: -4.711em; left: 1.13em;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 2.923em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.822em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.232em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.642em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 2.103em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 2.513em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(1.796em 1001.18em 4.562em -999.997em); top: -3.481em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1008.35em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.353em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1633" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="mi" id="MathJax-Span-1634" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1635" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-1636" style="font-family: STIXGeneral;">0</span><span class="mo" id="MathJax-Span-1637" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-1638" style="font-family: STIXGeneral; padding-left: 0.207em;">1</span><span class="mo" id="MathJax-Span-1639" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.997em; border-left: 0px solid; width: 0px; height: 5.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mover><mi>X</mi><mo accent="false">¯</mo></mover><mo>−</mo><mover><mi>Y</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>x</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><msqrt><mfrac><msubsup><mi>σ</mi><mi>x</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>x</mi></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup><msub><mi>n</mi><mi>y</mi></msub></mfrac></msqrt></mfrac></mstyle><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-154">\dfrac{(\overline{X} - \overline{Y}) - (\mu_x - \mu_y)}{\sqrt{\frac{\sigma^2_x}{n_x} + \frac{\sigma^2_y}{n_y}}} \sim N(0, 1)</script></span> <br> Note that for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-155-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>x</mi></msub><mo>&gt;</mo><mn>30</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1640" style="width: 3.896em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.179em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.18em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1641"><span class="msubsup" id="MathJax-Span-1642"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1643" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1644" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1645" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-1646" style="font-family: STIXGeneral; padding-left: 0.31em;">30</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>x</mi></msub><mo>></mo><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-155">n_x > 30</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-156-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>y</mi></msub><mo>&gt;</mo><mn>30</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1647" style="width: 3.896em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.179em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.18em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1648"><span class="msubsup" id="MathJax-Span-1649"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1650" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1651" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1652" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-1653" style="font-family: STIXGeneral; padding-left: 0.31em;">30</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mi>y</mi></msub><mo>></mo><mn>30</mn></math></span></span><script type="math/tex" id="MathJax-Element-156">n_y > 30</script></span> the requirement for normal populations canbe relaxed and/or sample variances can be used as estimates of the population variances</li>
<li>To test hypotheses about the equality of (or given difference in) two normal population means, based upon independent random samples and unknown but equal population variances use <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-157-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mover><mi>X</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x2212;</mo><mover><mi>Y</mi><mo accent="false">&#x00AF;</mo></mover><mo stretchy="false">)</mo><mo>&#x2212;</mo><mo stretchy="false">(</mo><msub><mi>&#x03BC;</mi><mi>x</mi></msub><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><msqrt><msubsup><mi>s</mi><mi>p</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><msub><mi>n</mi><mi>x</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>n</mi><mi>y</mi></msub></mfrac></mrow><mo>)</mo></mrow></msqrt></mfrac></mstyle><mo>&#x223C;</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mi>x</mi></msub><mo>+</mo><msub><mi>n</mi><mi>y</mi></msub><mo>,</mo><mtext>&#xA0;</mtext><mo>&#x2212;</mo><mn>2</mn></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1654" style="width: 16.089em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.169em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.617em 1013.17em 5.126em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1655"><span class="mstyle" id="MathJax-Span-1656"><span class="mrow" id="MathJax-Span-1657"><span class="mfrac" id="MathJax-Span-1658"><span style="display: inline-block; position: relative; width: 8.353em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.82em 1008.2em 4.46em -999.997em); top: -4.762em; left: 50%; margin-left: -4.096em;"><span class="mrow" id="MathJax-Span-1659"><span class="mo" id="MathJax-Span-1660" style="font-family: STIXGeneral;">(</span><span class="munderover" id="MathJax-Span-1661"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1662" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-1663" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1664" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="munderover" id="MathJax-Span-1665" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1666" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.67em 3.998em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-1667" style=""><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.412em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.31em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1668" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1669" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mo" id="MathJax-Span-1670" style="font-family: STIXGeneral; padding-left: 0.259em;">(</span><span class="msubsup" id="MathJax-Span-1671"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1672" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1673" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1674" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-1675" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1676" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1677" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1678" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.359em 1006.56em 5.126em -999.997em); top: -2.508em; left: 50%; margin-left: -3.276em;"><span class="msqrt" id="MathJax-Span-1679"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px;"><span style="position: absolute; clip: rect(2.666em 1005.28em 4.818em -999.997em); top: -3.993em; left: 1.13em;"><span class="mrow" id="MathJax-Span-1680"><span class="msubsup" id="MathJax-Span-1681"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1682" style="font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.412em;"><span class="mn" id="MathJax-Span-1683" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.306em -999.997em); top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-1684" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mrow" id="MathJax-Span-1685" style="padding-left: 0.207em;"><span class="mo" id="MathJax-Span-1686" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">(</span></span><span class="mrow" id="MathJax-Span-1687"><span class="mfrac" id="MathJax-Span-1688"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1689" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.62em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1690"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1691" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1692" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.77em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.771em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1693" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mfrac" id="MathJax-Span-1694" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1695" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.62em 4.357em -999.997em); top: -3.635em; left: 50%; margin-left: -0.305em;"><span class="msubsup" id="MathJax-Span-1696"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1697" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1698" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.77em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.771em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span class="mo" id="MathJax-Span-1699" style="vertical-align: -0.407em;"><span style="font-family: STIXSizeTwoSym;">)</span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1005.43em 3.384em -999.997em); top: -4.659em; left: 1.13em;"><span style="display: inline-block; position: relative; width: 5.433em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 4.921em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.873em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.335em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.796em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 2.205em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 2.666em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 3.128em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 3.589em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 4.05em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 4.46em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(1.796em 1001.18em 4.562em -999.997em); top: -3.43em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1008.35em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.353em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1700" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1701" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.281em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1702" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.259em;"><span class="texatom" id="MathJax-Span-1703"><span class="mrow" id="MathJax-Span-1704"><span class="msubsup" id="MathJax-Span-1705"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1706" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1707" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1708" style="font-size: 70.7%; font-family: STIXGeneral;">+</span><span class="msubsup" id="MathJax-Span-1709"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1710" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1711" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1712" style="font-size: 70.7%; font-family: STIXGeneral;">,</span><span class="mtext" id="MathJax-Span-1713" style="font-size: 70.7%; font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1714" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1715" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.997em; border-left: 0px solid; width: 0px; height: 5.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><mover><mi>X</mi><mo accent="false">¯</mo></mover><mo>−</mo><mover><mi>Y</mi><mo accent="false">¯</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>x</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow><msqrt><msubsup><mi>s</mi><mi>p</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><msub><mi>n</mi><mi>x</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>n</mi><mi>y</mi></msub></mfrac></mrow><mo>)</mo></mrow></msqrt></mfrac></mstyle><mo>∼</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>n</mi><mi>x</mi></msub><mo>+</mo><msub><mi>n</mi><mi>y</mi></msub><mo>,</mo><mtext> </mtext><mo>−</mo><mn>2</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-157">\dfrac{(\overline{X} - \overline{Y}) - (\mu_x -\mu_y)}{\sqrt{s_p^2\left(\frac{1}{n_x} + \frac{1}{n_y} \right)}} \sim t_{n_x + n_y,\ -2}</script></span> <br> where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-158-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>s</mi><mi>p</mi><mn>2</mn></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>n</mi><mi>x</mi></msub><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>s</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mi>y</mi></msub><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>s</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mrow><msub><mi>n</mi><mi>x</mi></msub><mo>+</mo><msub><mi>n</mi><mi>y</mi></msub><mo>&#x2212;</mo><mn>2</mn></mrow></mfrac></mstyle></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1716" style="width: 14.449em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.837em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1011.84em 3.691em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1717"><span class="msubsup" id="MathJax-Span-1718"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1719" style="font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.412em;"><span class="mn" id="MathJax-Span-1720" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.306em -999.997em); top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-1721" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1722" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mstyle" id="MathJax-Span-1723" style="padding-left: 0.31em;"><span class="mrow" id="MathJax-Span-1724"><span class="mfrac" id="MathJax-Span-1725"><span style="display: inline-block; position: relative; width: 9.48em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1009.38em 4.46em -999.997em); top: -4.762em; left: 50%; margin-left: -4.659em;"><span class="mrow" id="MathJax-Span-1726"><span class="mo" id="MathJax-Span-1727" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1728"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1729" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1730" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1731" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1732" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-1733" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1734"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1735" style="font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.412em;"><span class="mn" id="MathJax-Span-1736" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-1737" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1738" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mo" id="MathJax-Span-1739" style="font-family: STIXGeneral; padding-left: 0.259em;">(</span><span class="msubsup" id="MathJax-Span-1740"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1741" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1742" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1743" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1744" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-1745" style="font-family: STIXGeneral;">)</span><span class="msubsup" id="MathJax-Span-1746"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1747" style="font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.352em; left: 0.412em;"><span class="mn" id="MathJax-Span-1748" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.306em -999.997em); top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-1749" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1004.66em 4.46em -999.997em); top: -3.327em; left: 50%; margin-left: -2.354em;"><span class="mrow" id="MathJax-Span-1750"><span class="msubsup" id="MathJax-Span-1751"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1752" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1753" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1754" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-1755" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1756" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1757" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1758" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1759" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1009.48em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 9.48em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.247em; border-left: 0px solid; width: 0px; height: 3.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>s</mi><mi>p</mi><mn>2</mn></msubsup><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>n</mi><mi>x</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>s</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><msub><mi>n</mi><mi>y</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>s</mi><mi>y</mi><mn>2</mn></msubsup></mrow><mrow><msub><mi>n</mi><mi>x</mi></msub><mo>+</mo><msub><mi>n</mi><mi>y</mi></msub><mo>−</mo><mn>2</mn></mrow></mfrac></mstyle></math></span></span><script type="math/tex" id="MathJax-Element-158">s_p^2 = \dfrac{(n_x-1)s_x^2 + (n_y-1)s_y^2}{n_x + n_y - 2}</script></span></li>
</ol><h2 id="testing-for-goodness-of-fit"><a class="anchor hidden-xs" href="#testing-for-goodness-of-fit" title="testing-for-goodness-of-fit"><span class="octicon octicon-link"></span></a>Testing for goodness of fit</h2><ol>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-159-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2211;</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>O</mi><mo>&#x2212;</mo><mi>E</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>E</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1760" style="width: 4.511em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.691em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1003.69em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1761"><span class="mo" id="MathJax-Span-1762" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="mfrac" id="MathJax-Span-1763" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 2.359em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.23em 1002.21em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -1.124em;"><span class="mrow" id="MathJax-Span-1764"><span class="mo" id="MathJax-Span-1765" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1766" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">O</span><span class="mo" id="MathJax-Span-1767" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-1768" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1769"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.21em 4.255em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1770" style="font-size: 70.7%; font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.259em;"><span class="mn" id="MathJax-Span-1771" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1772" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.36em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.359em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 2.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>O</mi><mo>−</mo><mi>E</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mi>E</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-159">\sum \frac{(O-E)^2}{E}</script></span> may be approximated by a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-160-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C7;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1773" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1000.98em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1774"><span class="msubsup" id="MathJax-Span-1775"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.51em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1776" style="font-family: STIXGeneral; font-style: italic;">χ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.566em;"><span class="mn" id="MathJax-Span-1777" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-160">\chi^2</script></span> distribution provided that
<ul>
<li>The <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-161-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1778" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1779"><span class="mi" id="MathJax-Span-1780" style="font-family: STIXGeneral; font-style: italic;">O</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math></span></span><script type="math/tex" id="MathJax-Element-161">O</script></span>s are frequenceies,</li>
<li>The <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-162-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1781" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1782"><span class="mi" id="MathJax-Span-1783" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-162">E</script></span>s are at least five,</li>
<li>The classes form a sample space that is, every possible observation fits into one and only one class</li>
</ul>
</li>
<li>The number of degrees of freedom is the number of classes, minus the number of independent pieces of information derived from the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-163-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1784" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1785"><span class="mi" id="MathJax-Span-1786" style="font-family: STIXGeneral; font-style: italic;">O</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math></span></span><script type="math/tex" id="MathJax-Element-163">O</script></span>s in order to calculate the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-164-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1787" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1788"><span class="mi" id="MathJax-Span-1789" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math></span></span><script type="math/tex" id="MathJax-Element-164">E</script></span>s</li>
<li>If there are <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-165-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1790" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1791"><span class="mi" id="MathJax-Span-1792" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-165">k</script></span> classes and any necessary parameters are estimated from the data the number of degrees of is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-166-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#x2212;</mo><mn>2</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1793" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1794"><span class="mi" id="MathJax-Span-1795" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1796" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1797" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-166">k-2</script></span> for a Poisson, binomial, or exponential distribution, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-167-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#x2212;</mo><mn>3</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1798" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1799"><span class="mi" id="MathJax-Span-1800" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1801" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1802" style="font-family: STIXGeneral; padding-left: 0.259em;">3</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-167">k-3</script></span> for a normal distribution</li>
</ol><h1 id="ss06"><a class="anchor hidden-xs" href="#ss06" title="ss06"><span class="octicon octicon-link"></span></a>SS06</h1><h2 id="experimental-design"><a class="anchor hidden-xs" href="#experimental-design" title="experimental-design"><span class="octicon octicon-link"></span></a>Experimental design</h2><ol>
<li>Experimental error is the effect of factors other than those controlled by the experimenter</li>
<li>In a paired comparison, experimental error is reduced by applying both treatments to the same subjects or in the same conditions</li>
<li>The purpose of randomisation is to eliminate bias</li>
<li>Blocking is used to reduce experimental error by applying treatments (usually more than two) to the same subjects or in the same conditions</li>
<li>If a new treatement is applied to an experimental group, a control group, which receives no treatment or the standard treatment, is needed to act as a measure of the effect of not applying the new treatment</li>
<li>A placebo is a pill or treatment which contains no active ingredient</li>
<li>In a blind trial subjects do not know whether they are receiving the treatment or a placebo</li>
<li>In a double blind trial neither the subject nor the person administering the treatment knows whether a placebo or an active drug is being given</li>
</ol><h2 id="analysis-of-paired-comparisons"><a class="anchor hidden-xs" href="#analysis-of-paired-comparisons" title="analysis-of-paired-comparisons"><span class="octicon octicon-link"></span></a>Analysis of paired comparisons</h2><p>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-168-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>D</mi><mo accent="false">&#x00AF;</mo></mover></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1803" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1000.77em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1804"><span class="munderover" id="MathJax-Span-1805"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1806" style="font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.77em 3.998em -999.997em); top: -4.711em; left: 0em;"><span class="mo" id="MathJax-Span-1807" style=""><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.515em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.361em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>D</mi><mo accent="false">¯</mo></mover></math></span></span><script type="math/tex" id="MathJax-Element-168">\overline{D}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-169-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>d</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1808" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1809"><span class="msubsup" id="MathJax-Span-1810"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1811" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1812" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>d</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-169">S_d</script></span> denote the mean and standard deviation, respectively, of a random sample of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-170-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1813" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1814"><span class="mi" id="MathJax-Span-1815" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-170">n</script></span> differences that can be assumed to be bormally distributed with mean <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-171-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03BC;</mi><mi>d</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1816" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.642em 1000.98em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1817"><span class="msubsup" id="MathJax-Span-1818"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1819" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1820" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>d</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-171">\mu_d</script></span> then <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-172-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mover><mi>D</mi><mo accent="false">&#x00AF;</mo></mover><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>d</mi></msub></mrow><mfrac><msub><mi>S</mi><mi>d</mi></msub><msqrt><mi>n</mi></msqrt></mfrac></mfrac></mstyle><mo>&#x223C;</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1821" style="width: 7.431em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.099em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.72em 1006.1em 4.255em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1822"><span class="mstyle" id="MathJax-Span-1823"><span class="mrow" id="MathJax-Span-1824"><span class="mfrac" id="MathJax-Span-1825"><span style="display: inline-block; position: relative; width: 3.025em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.82em 1002.92em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.483em;"><span class="mrow" id="MathJax-Span-1826"><span class="munderover" id="MathJax-Span-1827"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0.003em;"><span class="mi" id="MathJax-Span-1828" style="font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.77em 3.998em -999.997em); top: -4.711em; left: 0em;"><span class="mo" id="MathJax-Span-1829" style=""><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; top: -3.993em; left: 0em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.515em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.156em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.993em; left: 0.361em;"><span style="font-size: 70.7%; font-family: STIXGeneral;">⎯</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1830" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-1831" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1832" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1833" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.871em 1001.23em 4.716em -999.997em); top: -3.02em; left: 50%; margin-left: -0.612em;"><span class="mfrac" id="MathJax-Span-1834"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.67em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.356em;"><span class="msubsup" id="MathJax-Span-1835"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1836" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1837" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.357em -999.997em); top: -3.584em; left: 50%; margin-left: -0.459em;"><span class="msqrt" id="MathJax-Span-1838"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-1839"><span class="mi" id="MathJax-Span-1840" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.36em 1.283em -999.997em); top: -1.586em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.788em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.98em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.976em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.02em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.025em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span class="mo" id="MathJax-Span-1841" style="font-family: STIXGeneral; padding-left: 0.31em;">∼</span><span class="msubsup" id="MathJax-Span-1842" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1843" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.259em;"><span class="texatom" id="MathJax-Span-1844"><span class="mrow" id="MathJax-Span-1845"><span class="mi" id="MathJax-Span-1846" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1847" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1848" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.934em; border-left: 0px solid; width: 0px; height: 4.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mover><mi>D</mi><mo accent="false">¯</mo></mover><mo>−</mo><msub><mi>μ</mi><mi>d</mi></msub></mrow><mfrac><msub><mi>S</mi><mi>d</mi></msub><msqrt><mi>n</mi></msqrt></mfrac></mfrac></mstyle><mo>∼</mo><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-172">\dfrac{\overline{D}-\mu_d}{\frac{S_d}{\sqrt{n}}} \sim t_{n-1}</script></span></p><h2 id="analysis-of-variance-anova"><a class="anchor hidden-xs" href="#analysis-of-variance-anova" title="analysis-of-variance-anova"><span class="octicon octicon-link"></span></a>Analysis of variance (ANOVA)</h2><ol>
<li>
<p>The assumptions for the three models considered, one and two factor ANOVAs, and Latin square designs, are that:<br>
a. The observations are obtained independently and randomly from populations at each factor level (combination)<br>
b. These populations are (approximately) normally distributed with common variance <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-173-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1849" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1001.03em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1850"><span class="msubsup" id="MathJax-Span-1851"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1852" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-1853" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-173">\sigma^2</script></span><br>
c. When two or more factors are involved, there is no interaction between them</p>
</li>
<li>
<p>One way ANOVA table</p>
<table>
<thead>
<tr>
<th>Source of variation</th>
<th>Sum of squares</th>
<th>Degrees of freedom</th>
<th>Mean square</th>
<th>F ratio</th>
</tr>
</thead>
<tbody>
<tr>
<td>Between samples</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-174-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>B</mi><mo>&#x2217;</mo></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1854" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.54em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1855"><span class="mi" id="MathJax-Span-1856" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1857"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1858" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.46em 4.152em -999.997em); top: -4.301em; left: 0.566em;"><span class="mo" id="MathJax-Span-1859" style="font-size: 70.7%; font-family: STIXGeneral;">∗</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-1860" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>B</mi><mo>∗</mo></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-174">SS_B^*</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-175-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1861" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.05em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1862"><span class="mi" id="MathJax-Span-1863" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1864" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1865" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-175">k-1</script></span></td>
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position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1870" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1871" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">w</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1872" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1873" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1874"><span class="mi" id="MathJax-Span-1875" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1876"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1877" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1878" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -3.584em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-1879"><span class="mi" id="MathJax-Span-1880" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1881" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1882" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.28em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.283em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.684em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>w</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>B</mi></msub></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-176">MS_w = \frac{SS_B}{k-1}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-177-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msub><mi>S</mi><mi>B</mi></msub></mrow><mrow><mi>M</mi><msub><mi>S</mi><mi>W</mi></msub></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1883" style="width: 2.205em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1001.8em 2.871em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1884"><span class="mfrac" id="MathJax-Span-1885"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.34em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.663em;"><span class="mrow" id="MathJax-Span-1886"><span class="mi" id="MathJax-Span-1887" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1888"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1889" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1890" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.49em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.715em;"><span class="mrow" id="MathJax-Span-1891"><span class="mi" id="MathJax-Span-1892" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1893"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1894" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1895" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.59em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.591em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.684em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msub><mi>S</mi><mi>B</mi></msub></mrow><mrow><mi>M</mi><msub><mi>S</mi><mi>W</mi></msub></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-177">\frac{MS_B}{MS_W}</script></span></td>
</tr>
<tr>
<td>Within samples</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-178-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msub><mi>S</mi><mi>W</mi></msub><mo>=</mo><mi>S</mi><msub><mi>S</mi><mi>T</mi></msub><mo>&#x2212;</mo><mi>S</mi><msub><mi>S</mi><mi>B</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1896" style="width: 8.916em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.277em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1007.28em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1897"><span class="mi" id="MathJax-Span-1898" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1899"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1900" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1901" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1902" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1903" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1904"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1905" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1906" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1907" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1908" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1909"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1910" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1911" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msub><mi>S</mi><mi>W</mi></msub><mo>=</mo><mi>S</mi><msub><mi>S</mi><mi>T</mi></msub><mo>−</mo><mi>S</mi><msub><mi>S</mi><mi>B</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-178">SS_W = SS_T - SS_B</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-179-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mi>k</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1912" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.15em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1913"><span class="mi" id="MathJax-Span-1914" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1915" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-1916" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-179">n-k</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-180-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>W</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>W</mi></msub></mrow><mrow><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1917" style="width: 6.099em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.972em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.437em 1004.97em 3.179em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1918"><span class="mi" id="MathJax-Span-1919" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1920"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1921" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1922" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1923" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1924" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.23em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-1925"><span class="mi" id="MathJax-Span-1926" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1927"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1928" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1929" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-1930"><span class="mi" id="MathJax-Span-1931" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1932" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1933" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>W</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>W</mi></msub></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-180">MS_W = \frac{SS_W}{n-1}</script></span></td>
<td></td>
</tr>
<tr>
<td><strong>Total</strong></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-181-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>T</mi><mo>&#x2217;</mo></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1934" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.54em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1935"><span class="mi" id="MathJax-Span-1936" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1937"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1938" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.46em 4.152em -999.997em); top: -4.301em; left: 0.566em;"><span class="mo" id="MathJax-Span-1939" style="font-size: 70.7%; font-family: STIXGeneral;">∗</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-1940" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>T</mi><mo>∗</mo></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-181">SS_T^*</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-182-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1941" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1942"><span class="mi" id="MathJax-Span-1943" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1944" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1945" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-182">n-1</script></span></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
</li>
<li>
<p>Two way ANOVA table</p>
<table>
<thead>
<tr>
<th>Source of variation</th>
<th>Sum of squares</th>
<th>Degrees of freedom</th>
<th>Mean square</th>
<th>F ratio</th>
</tr>
</thead>
<tbody>
<tr>
<td>Between rows</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-183-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>B</mi><mo>&#x2217;</mo></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1946" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.54em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1947"><span class="mi" id="MathJax-Span-1948" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1949"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1950" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.46em 4.152em -999.997em); top: -4.301em; left: 0.566em;"><span class="mo" id="MathJax-Span-1951" style="font-size: 70.7%; font-family: STIXGeneral;">∗</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-1952" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">B</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>B</mi><mo>∗</mo></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-183">SS_B^*</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-184-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1953" style="width: 2.974em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.31em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1954"><span class="mi" id="MathJax-Span-1955" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mo" id="MathJax-Span-1956" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1957" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-184">m-1</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-185-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>R</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>R</mi></msub></mrow><mrow><mi>m</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1958" style="width: 5.996em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.921em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.437em 1004.92em 3.179em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1959"><span class="mi" id="MathJax-Span-1960" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1961"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1962" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-1963" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1964" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1965" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.488em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-1966"><span class="mi" id="MathJax-Span-1967" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1968"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1969" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1970" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.28em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.663em;"><span class="mrow" id="MathJax-Span-1971"><span class="mi" id="MathJax-Span-1972" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">m</span><span class="mo" id="MathJax-Span-1973" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1974" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.49em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.488em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>R</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>R</mi></msub></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-185">MS_R = \frac{SS_R}{m-1}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-186-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msub><mi>S</mi><mi>R</mi></msub></mrow><mrow><mi>M</mi><msub><mi>S</mi><mi>E</mi></msub></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1975" style="width: 2.103em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.693em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1001.69em 2.82em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1976"><span class="mfrac" id="MathJax-Span-1977"><span style="display: inline-block; position: relative; width: 1.488em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.34em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.663em;"><span class="mrow" id="MathJax-Span-1978"><span class="mi" id="MathJax-Span-1979" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1980"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1981" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1982" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.34em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.663em;"><span class="mrow" id="MathJax-Span-1983"><span class="mi" id="MathJax-Span-1984" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1985"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1986" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-1987" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.49em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.488em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>M</mi><msub><mi>S</mi><mi>R</mi></msub></mrow><mrow><mi>M</mi><msub><mi>S</mi><mi>E</mi></msub></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-186">\frac{MS_R}{MS_E}</script></span></td>
</tr>
<tr>
<td>Between columns</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-187-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>C</mi><mo>&#x2217;</mo></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1988" style="width: 1.949em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.59em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1989"><span class="mi" id="MathJax-Span-1990" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1991"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1992" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.46em 4.152em -999.997em); top: -4.301em; left: 0.566em;"><span class="mo" id="MathJax-Span-1993" style="font-size: 70.7%; font-family: STIXGeneral;">∗</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.57em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-1994" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>C</mi><mo>∗</mo></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-187">SS_C^*</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-188-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1995" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1996"><span class="mi" id="MathJax-Span-1997" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1998" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-1999" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-188">n-1</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-189-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>C</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>C</mi></msub></mrow><mrow><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2000" style="width: 5.894em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.818em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.437em 1004.82em 3.179em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2001"><span class="mi" id="MathJax-Span-2002" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-2003"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2004" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2005" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2006" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2007" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.13em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-2008"><span class="mi" id="MathJax-Span-2009" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2010"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2011" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-2012" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-2013"><span class="mi" id="MathJax-Span-2014" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2015" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2016" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>C</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>C</mi></msub></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-189">MS_C = \frac{SS_C}{n-1}</script></span></td>
<td>$$</td>
</tr>
<tr>
<td>Error</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-190-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msub><mi>S</mi><mi>E</mi></msub><mo>=</mo><mi>S</mi><msub><mi>S</mi><mi>T</mi></msub><mo>&#x2212;</mo><mi>S</mi><msub><mi>S</mi><mi>R</mi></msub><mo>&#x2212;</mo><mi>S</mi><msub><mi>S</mi><mi>C</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2017" style="width: 12.041em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.839em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1009.84em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2018"><span class="mi" id="MathJax-Span-2019" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2020"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2021" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2022" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2023" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2024" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2025"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2026" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2027" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2028" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2029" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2030"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2031" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2032" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2033" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2034" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2035"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2036" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2037" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msub><mi>S</mi><mi>E</mi></msub><mo>=</mo><mi>S</mi><msub><mi>S</mi><mi>T</mi></msub><mo>−</mo><mi>S</mi><msub><mi>S</mi><mi>R</mi></msub><mo>−</mo><mi>S</mi><msub><mi>S</mi><mi>C</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-190">SS_E = SS_T - SS_R - SS_C</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-191-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>m</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2038" style="width: 7.277em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.945em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.89em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2039"><span class="mo" id="MathJax-Span-2040" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2041" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mo" id="MathJax-Span-2042" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2043" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2044" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2045" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2046" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2047" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2048" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2049" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-191">(m-1)(n-1)</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-192-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>E</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>E</mi></msub></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2050" style="width: 8.558em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.021em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.437em 1007.02em 3.23em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2051"><span class="mi" id="MathJax-Span-2052" style="font-family: STIXGeneral; font-style: italic;">M<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-2053"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2054" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2055" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2056" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2057" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -4.506em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-2058"><span class="mi" id="MathJax-Span-2059" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2060"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2061" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.891em; left: 0.361em;"><span class="mi" id="MathJax-Span-2062" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1003.43em 4.255em -999.997em); top: -3.635em; left: 50%; margin-left: -1.739em;"><span class="mrow" id="MathJax-Span-2063"><span class="mo" id="MathJax-Span-2064" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2065" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">m</span><span class="mo" id="MathJax-Span-2066" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2067" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2068" style="font-size: 70.7%; font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2069" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2070" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2071" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2072" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2073" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.59em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.589em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.684em; border-left: 0px solid; width: 0px; height: 1.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><msub><mi>S</mi><mi>E</mi></msub><mo>=</mo><mfrac><mrow><mi>S</mi><msub><mi>S</mi><mi>E</mi></msub></mrow><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-192">MS_E = \frac{SS_E}{(m-1)(n-1)}</script></span></td>
<td></td>
</tr>
<tr>
<td><strong>Total</strong></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-193-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>T</mi><mo>&#x2217;</mo></msubsup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2074" style="width: 1.898em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.54em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2075"><span class="mi" id="MathJax-Span-2076" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2077"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2078" style="font-family: STIXGeneral; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.46em 4.152em -999.997em); top: -4.301em; left: 0.566em;"><span class="mo" id="MathJax-Span-2079" style="font-size: 70.7%; font-family: STIXGeneral;">∗</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.686em; left: 0.515em;"><span class="mi" id="MathJax-Span-2080" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><msubsup><mi>S</mi><mi>T</mi><mo>∗</mo></msubsup></math></span></span><script type="math/tex" id="MathJax-Element-193">SS_T^*</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-194-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2081" style="width: 3.589em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2082"><span class="mi" id="MathJax-Span-2083" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-2084" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2085" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2086" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>n</mi><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-194">mn -1</script></span></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
</li>
</ol><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-195-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2217;</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2087" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.666em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2088"><span class="mo" id="MathJax-Span-2089" style="font-family: STIXGeneral;">∗</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: 0.003em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∗</mo></math></span></span><script type="math/tex" id="MathJax-Element-195">*</script></span> Provided in the formulae booklet</p><h2 id="statistical-process-control"><a class="anchor hidden-xs" href="#statistical-process-control" title="statistical-process-control"><span class="octicon octicon-link"></span></a>Statistical process control</h2><ol>
<li>Statistical process control may be used when a large number of similar items are being produced. Its purpose is to give a signal when the process mean has moved away from the target value or when item-to-item variability has increased</li>
<li>For control charts for means:
<ul>
<li>Sample mean between warning limits- No action</li>
<li>Sample mean between arning and action limits- Take another sample immediately. If new sample mean outside warning limits take action</li>
<li>Sample mean outside action limits- Take action</li>
</ul>
</li>
<li>The warning limits are set at <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-196-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi><mo>&#x00B1;</mo><mn>1.96</mn><mfrac><mi>&#x03C3;</mi><msqrt><mi>n</mi></msqrt></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2090" style="width: 5.689em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.664em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1004.66em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2091"><span class="mi" id="MathJax-Span-2092" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-2093" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mn" id="MathJax-Span-2094" style="font-family: STIXGeneral; padding-left: 0.259em;">1.96</span><span class="mfrac" id="MathJax-Span-2095"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2096" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.357em -999.997em); top: -3.584em; left: 50%; margin-left: -0.459em;"><span class="msqrt" id="MathJax-Span-2097"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-2098"><span class="mi" id="MathJax-Span-2099" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.36em 1.283em -999.997em); top: -1.586em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.788em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.98em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.976em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>±</mo><mn>1.96</mn><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-196">\mu \pm 1.96\frac{\sigma}{\sqrt{n}}</script></span>, and the action limits at <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-197-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi><mo>&#x00B1;</mo><mn>3.09</mn><mfrac><mi>&#x03C3;</mi><msqrt><mi>n</mi></msqrt></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2100" style="width: 5.689em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.664em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1004.66em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2101"><span class="mi" id="MathJax-Span-2102" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-2103" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mn" id="MathJax-Span-2104" style="font-family: STIXGeneral; padding-left: 0.259em;">3.09</span><span class="mfrac" id="MathJax-Span-2105"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2106" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.357em -999.997em); top: -3.584em; left: 50%; margin-left: -0.459em;"><span class="msqrt" id="MathJax-Span-2107"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0.515em;"><span class="mrow" id="MathJax-Span-2108"><span class="mi" id="MathJax-Span-2109" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.976em 1000.36em 1.283em -999.997em); top: -1.586em; left: 0.515em;"><span style="display: inline-block; overflow: hidden; vertical-align: -0.049em; border-top: 1.2px solid; width: 0.361em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.788em; left: 0em;"><span><span style="font-size: 70.7%; font-family: STIXVariants;">√</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.98em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.976em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>±</mo><mn>3.09</mn><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-197">\mu \pm 3.09\frac{\sigma}{\sqrt{n}}</script></span>, where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-198-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2110" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2111"><span class="mi" id="MathJax-Span-2112" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-198">\mu</script></span> is the target value, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-199-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2113" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2114"><span class="mi" id="MathJax-Span-2115" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-199">\sigma</script></span> is the short-term standard deviation, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-200-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2116" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2117"><span class="mi" id="MathJax-Span-2118" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-200">n</script></span> is the sample size</li>
<li>Variability may be controlled by plotting the sample ranges or standard deviations on control charts. The limits for these charts are found by multiplying the process short-term standard deviation found by factors in the control charts for variability (Table 12)</li>
<li>When the standard deviation must be estimated from a number of small samples the average sample range can be calculated and a factor from table 12 applied. <br> Alternatively <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-201-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>i</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2119" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.67em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2120"><span class="msubsup" id="MathJax-Span-2121"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2122" style="font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.412em;"><span class="mi" id="MathJax-Span-2123" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>i</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-201">s_i</script></span> can be calculated for each sample and the formula <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-202-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msqrt><mfrac><mrow><mo>&#x2211;</mo><msubsup><mi>s</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mi>n</mi></mfrac></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2124" style="width: 5.587em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.562em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1004.56em 3.179em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2125"><span class="mi" id="MathJax-Span-2126" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-2127" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msqrt" id="MathJax-Span-2128" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px;"><span style="position: absolute; clip: rect(2.666em 1001.8em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-2129"><span class="mfrac" id="MathJax-Span-2130"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.281em 1001.44em 4.357em -999.997em); top: -4.608em; left: 50%; margin-left: -0.715em;"><span class="mrow" id="MathJax-Span-2131"><span class="mo" id="MathJax-Span-2132" style="font-size: 70.7%; font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span class="msubsup" id="MathJax-Span-2133" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px;"><span style="position: absolute; clip: rect(3.537em 1000.26em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2134" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.486em 1000.31em 4.152em -999.997em); top: -4.249em; left: 0.259em;"><span class="mn" id="MathJax-Span-2135" style="font-size: 50%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.21em 4.152em -999.997em); top: -3.788em; left: 0.259em;"><span class="mi" id="MathJax-Span-2136" style="font-size: 50%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-2137" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.54em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.539em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.8em 3.384em -999.997em); top: -4.506em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.335em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.412em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.873em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.788em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><msqrt><mfrac><mrow><mo>∑</mo><msubsup><mi>s</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mi>n</mi></mfrac></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-202">s = \sqrt{\frac{\sum s_i^2}{n}}</script></span> evaluated</li>
<li>If the tolerance width exceeds six standard deviations the process should be able to meet the tolerances consistently, provided the mean is kept on target</li>
<li>For charts for proportion non-conforming providing <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-203-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2138" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2139"><span class="mi" id="MathJax-Span-2140" style="font-family: STIXGeneral; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-203">n</script></span> is reasonably large:
<ul>
<li>The warning limits are <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-204-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#x00B1;</mo><mn>1.96</mn><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2141" style="width: 8.455em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.919em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1006.92em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2142"><span class="mi" id="MathJax-Span-2143" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2144" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mn" id="MathJax-Span-2145" style="font-family: STIXGeneral; padding-left: 0.259em;">1.96</span><span class="msqrt" id="MathJax-Span-2146"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px;"><span style="position: absolute; clip: rect(2.82em 1002.36em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-2147"><span class="mfrac" id="MathJax-Span-2148"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-2149"><span class="mi" id="MathJax-Span-2150" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2151" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-2152" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2153" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2154" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2155" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-2156" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.154em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.41em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.898em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.361em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.771em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.13em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.539em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>±</mo><mn>1.96</mn><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-204">p \pm 1.96\sqrt{\frac{p(1-p)}{n}}</script></span></li>
<li>The action limits are <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-205-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#x00B1;</mo><mn>3.09</mn><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2157" style="width: 8.455em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.919em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1006.92em 3.281em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2158"><span class="mi" id="MathJax-Span-2159" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2160" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mn" id="MathJax-Span-2161" style="font-family: STIXGeneral; padding-left: 0.259em;">3.09</span><span class="msqrt" id="MathJax-Span-2162"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px;"><span style="position: absolute; clip: rect(2.82em 1002.36em 4.511em -999.997em); top: -3.993em; left: 1.078em;"><span class="mrow" id="MathJax-Span-2163"><span class="mfrac" id="MathJax-Span-2164"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -1.022em;"><span class="mrow" id="MathJax-Span-2165"><span class="mi" id="MathJax-Span-2166" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2167" style="font-size: 70.7%; font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-2168" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2169" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2170" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-2171" style="font-size: 70.7%; font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-2172" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.154em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.41em 3.384em -999.997em); top: -4.403em; left: 1.078em;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.898em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.361em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.771em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.13em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.539em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1001.08em 4.46em -999.997em); top: -3.737em; left: 0em;"><span style="font-family: STIXSizeOneSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.747em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>±</mo><mn>3.09</mn><msqrt><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><mi>n</mi></mfrac></msqrt></math></span></span><script type="math/tex" id="MathJax-Element-205">p \pm 3.09\sqrt{\frac{p(1-p)}{n}}</script></span></li>
</ul>
</li>
</ol><h2 id="acceptance-sampling"><a class="anchor hidden-xs" href="#acceptance-sampling" title="acceptance-sampling"><span class="octicon octicon-link"></span></a>Acceptance sampling</h2><ol>
<li>Acceptance sampling may be applied to large batches of similar items. It is the process of deciding whether or not the batch is acceptable by testing a small sample of the items</li>
<li>The operating characteristic for an acceptance sampling by attributes plan is a graph of probability of acceptance against proportion non-conforming in the batch</li>
<li>The probabilities may be found from the binomial distribution provided the sample is random and the sample size is small compared to the batch</li>
<li>In double sampling, the number of non-conforming items in the first sample will determine whether a decision is made immediately or whether it is delayed until a second sample has been inspected</li>
<li>For acceptance sampling by variables the operating characteristic is a graph of probability of acceptance against batch mean</li>
</ol><h1 id="ms03"><a class="anchor hidden-xs" href="#ms03" title="ms03"><span class="octicon octicon-link"></span></a>MS03</h1><h2 id="bayes’-theorem"><a class="anchor hidden-xs" href="#bayes’-theorem" title="bayes’-theorem"><span class="octicon octicon-link"></span></a>Bayes’ theorem</h2><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-206-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2173" style="width: 14.91em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.195em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.976em 1012.2em 3.794em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2174"><span class="mi" id="MathJax-Span-2175" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2176" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-2177"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2178" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2179" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="texatom" id="MathJax-Span-2180"><span class="mrow" id="MathJax-Span-2181"><span class="mo" id="MathJax-Span-2182" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2183" style="font-family: STIXGeneral; font-style: italic;">B</span><span class="mo" id="MathJax-Span-2184" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2185" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mstyle" id="MathJax-Span-2186" style="padding-left: 0.31em;"><span class="mrow" id="MathJax-Span-2187"><span class="mfrac" id="MathJax-Span-2188"><span style="display: inline-block; position: relative; width: 7.585em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1005.23em 4.46em -999.997em); top: -4.762em; left: 50%; margin-left: -2.61em;"><span class="mrow" id="MathJax-Span-2189"><span class="mi" id="MathJax-Span-2190" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2191" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-2192"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2193" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2194" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2195" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2196" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2197" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2198" style="font-family: STIXGeneral; font-style: italic;">B</span><span class="texatom" id="MathJax-Span-2199"><span class="mrow" id="MathJax-Span-2200"><span class="mo" id="MathJax-Span-2201" style="font-family: STIXVariants;">|</span></span></span><span class="msubsup" id="MathJax-Span-2202"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2203" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2204" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2205" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1007.43em 4.46em -999.997em); top: -3.225em; left: 50%; margin-left: -3.737em;"><span class="mrow" id="MathJax-Span-2206"><span class="munderover" id="MathJax-Span-2207"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px;"><span style="position: absolute; clip: rect(3.076em 1000.87em 4.408em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2208" style="font-family: STIXGeneral; vertical-align: 0.003em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.41em 4.152em -999.997em); top: -4.506em; left: 0.925em;"><span class="mi" id="MathJax-Span-2209" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.152em -999.997em); top: -3.686em; left: 0.925em;"><span class="texatom" id="MathJax-Span-2210"><span class="mrow" id="MathJax-Span-2211"><span class="mi" id="MathJax-Span-2212" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span class="mo" id="MathJax-Span-2213" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-2214" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-2215" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">P</span><span class="mo" id="MathJax-Span-2216" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-2217"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2218" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2219" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2220" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2221" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2222" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2223" style="font-family: STIXGeneral; font-style: italic;">B</span><span class="texatom" id="MathJax-Span-2224"><span class="mrow" id="MathJax-Span-2225"><span class="mo" id="MathJax-Span-2226" style="font-family: STIXVariants;">|</span></span></span><span class="msubsup" id="MathJax-Span-2227"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2228" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2229" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2230" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1007.59em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.585em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.372em; border-left: 0px solid; width: 0px; height: 3.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-206">P(A_j | B) = \dfrac{P(A_j)P(B|A_j)}{\sum_{i=1}^nP(A_i)P(B|A_i) }</script></span></p><p>Example</p><table>
<thead>
<tr>
<th>Event</th>
<th>P(A)</th>
<th>P(B)</th>
</tr>
</thead>
<tbody>
<tr>
<td>C</td>
<td>0.3</td>
<td>0.8</td>
</tr>
<tr>
<td>D</td>
<td>0.5</td>
<td>0.1</td>
</tr>
<tr>
<td>E</td>
<td>0.2</td>
<td>0.4</td>
</tr>
</tbody>
</table><p>After event A, event B occured. Find the probability that event E occured.</p><p>Let F be the probability that event B occured.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-207-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>E</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mn>0.2</mn><mo>&#x00D7;</mo><mn>0.4</mn></mrow><mrow><mn>0.3</mn><mo>&#x00D7;</mo><mn>0.8</mn><mo>+</mo><mn>0.5</mn><mo>&#x00D7;</mo><mn>0.1</mn><mo>+</mo><mn>0.2</mn><mo>&#x00D7;</mo><mn>0.4</mn></mrow></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mn>8</mn><mn>37</mn></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2231" style="width: 28.281em; display: inline-block;"><span style="display: inline-block; position: relative; width: 23.158em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-1.381em 1023em 5.996em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2232"><span class="mi" id="MathJax-Span-2233" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2234" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2235" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="texatom" id="MathJax-Span-2236"><span class="mrow" id="MathJax-Span-2237"><span class="mo" id="MathJax-Span-2238" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2239" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2240" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2241" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mtable" id="MathJax-Span-2242" style="padding-right: 0.156em; padding-left: 0.464em;"><span style="display: inline-block; position: relative; width: 18.701em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000em 8.712em -999.997em); top: -5.991em; left: 0em;"><span style="display: inline-block; position: relative; width: 0em; height: 0px;"><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -6.401em; right: 0em;"><span class="mtd" id="MathJax-Span-2243"><span class="mrow" id="MathJax-Span-2244"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -3.84em; right: 0em;"><span class="mtd" id="MathJax-Span-2302"><span class="mrow" id="MathJax-Span-2303"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -1.432em; right: 0em;"><span class="mtd" id="MathJax-Span-2325"><span class="mrow" id="MathJax-Span-2326"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.996em;"></span></span><span style="position: absolute; clip: rect(4.101em 1018.7em 11.478em -999.997em); top: -8.04em; left: 0em;"><span style="display: inline-block; position: relative; width: 18.701em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1018.7em 5.023em -999.997em); top: -6.401em; left: 0em;"><span class="mtd" id="MathJax-Span-2245"><span class="mrow" id="MathJax-Span-2246"><span class="mi" id="MathJax-Span-2247"></span><span class="mo" id="MathJax-Span-2248" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2249" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 17.164em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1004.77em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -2.405em;"><span class="mrow" id="MathJax-Span-2250"><span class="mi" id="MathJax-Span-2251" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2252" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2253" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2254" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2255" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2256" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2257" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="texatom" id="MathJax-Span-2258"><span class="mrow" id="MathJax-Span-2259"><span class="mo" id="MathJax-Span-2260" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2261" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2262" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1017.01em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -8.502em;"><span class="mrow" id="MathJax-Span-2263"><span class="mi" id="MathJax-Span-2264" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2265" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2266" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2267" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2268" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2269" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2270" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="texatom" id="MathJax-Span-2271"><span class="mrow" id="MathJax-Span-2272"><span class="mo" id="MathJax-Span-2273" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2274" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2275" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2276" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2277" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">P</span><span class="mo" id="MathJax-Span-2278" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2279" style="font-family: STIXGeneral; font-style: italic;">D</span><span class="mo" id="MathJax-Span-2280" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2281" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2282" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2283" style="font-family: STIXGeneral; font-style: italic;">D</span><span class="texatom" id="MathJax-Span-2284"><span class="mrow" id="MathJax-Span-2285"><span class="mo" id="MathJax-Span-2286" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2287" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2288" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2289" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2290" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">P</span><span class="mo" id="MathJax-Span-2291" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2292" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2293" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2294" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-2295" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2296" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="texatom" id="MathJax-Span-2297"><span class="mrow" id="MathJax-Span-2298"><span class="mo" id="MathJax-Span-2299" style="font-family: STIXVariants;">|</span></span></span><span class="mi" id="MathJax-Span-2300" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2301" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1017.16em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 17.164em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1015.06em 4.869em -999.997em); top: -3.84em; left: 0em;"><span class="mtd" id="MathJax-Span-2304"><span class="mrow" id="MathJax-Span-2305"><span class="mi" id="MathJax-Span-2306"></span><span class="mo" id="MathJax-Span-2307" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2308" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 13.527em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1003.64em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -1.842em;"><span class="mrow" id="MathJax-Span-2309"><span class="mn" id="MathJax-Span-2310" style="font-family: STIXGeneral;">0.2</span><span class="mo" id="MathJax-Span-2311" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-2312" style="font-family: STIXGeneral; padding-left: 0.259em;">0.4</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1013.37em 4.203em -999.997em); top: -3.327em; left: 50%; margin-left: -6.709em;"><span class="mrow" id="MathJax-Span-2313"><span class="mn" id="MathJax-Span-2314" style="font-family: STIXGeneral;">0.3</span><span class="mo" id="MathJax-Span-2315" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-2316" style="font-family: STIXGeneral; padding-left: 0.259em;">0.8</span><span class="mo" id="MathJax-Span-2317" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-2318" style="font-family: STIXGeneral; padding-left: 0.259em;">0.5</span><span class="mo" id="MathJax-Span-2319" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-2320" style="font-family: STIXGeneral; padding-left: 0.259em;">0.1</span><span class="mo" id="MathJax-Span-2321" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-2322" style="font-family: STIXGeneral; padding-left: 0.259em;">0.2</span><span class="mo" id="MathJax-Span-2323" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-2324" style="font-family: STIXGeneral; padding-left: 0.259em;">0.4</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1013.53em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 13.527em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1002.67em 4.869em -999.997em); top: -1.432em; left: 0em;"><span class="mtd" id="MathJax-Span-2327"><span class="mrow" id="MathJax-Span-2328"><span class="mi" id="MathJax-Span-2329"></span><span class="mo" id="MathJax-Span-2330" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2331" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2332" style="font-family: STIXGeneral;">8</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mn" id="MathJax-Span-2333" style="font-family: STIXGeneral;">37</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 8.046em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -4.059em; border-left: 0px solid; width: 0px; height: 8.753em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>E</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>F</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mn>0.2</mn><mo>×</mo><mn>0.4</mn></mrow><mrow><mn>0.3</mn><mo>×</mo><mn>0.8</mn><mo>+</mo><mn>0.5</mn><mo>×</mo><mn>0.1</mn><mo>+</mo><mn>0.2</mn><mo>×</mo><mn>0.4</mn></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>8</mn><mn>37</mn></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-207">P(E|F) = \begin{align}&= \frac{P(E)P(F|E)}{P(C)P(C|F) + P(D)P(D|F) + P(E)P(E|F)} \\ &= \frac{0.2\times0.4}{0.3\times0.8 + 0.5\times0.1 + 0.2\times0.4} \\&= \frac{8}{37} \end{align}</script></span></p><h2 id="linaer-combinations-of-random-variables"><a class="anchor hidden-xs" href="#linaer-combinations-of-random-variables" title="linaer-combinations-of-random-variables"><span class="octicon octicon-link"></span></a>Linaer combinations of random variables</h2><p>Covariance is a measure of the joint variability of two random variables.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-208-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><msub><mi>&#x03BC;</mi><mi>x</mi></msub><msub><mi>&#x03BC;</mi><mi>y</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2334" style="width: 25.003em; display: inline-block;"><span style="display: inline-block; position: relative; width: 20.494em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1020.49em 3.025em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2335"><span class="mi" id="MathJax-Span-2336" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2337" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-2338" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-2339" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2340" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2341" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-2342" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2343" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2344" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2345" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2346" style="font-family: STIXGeneral;">(</span><span class="mo" id="MathJax-Span-2347" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2348" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2349" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-2350" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2351" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2352" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2353" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2354" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2355" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2356" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-2357" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2358" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2359" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2360" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2361" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2362" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2363" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2364" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2365" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-2366" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2367" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2368" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-2369" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2370" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2371" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2372"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2373" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2374" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.434em; border-left: 0px solid; width: 0px; height: 1.316em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><msub><mi>μ</mi><mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>−</mo><msub><mi>μ</mi><mi>y</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>μ</mi><mi>x</mi></msub><msub><mi>μ</mi><mi>y</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-208">Cov(X, Y) = E((X - \mu_x)(Y - \mu_y)) = E(XY)-\mu_x\mu_y</script></span></p><p>The covariance can be used to find the product moment correlation coefficient of two random variables:</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-209-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03C1;</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>&#x03C3;</mi><mi>x</mi></msub><msub><mi>&#x03C3;</mi><mi>y</mi></msub></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2375" style="width: 7.533em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.15em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1006.15em 3.691em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2376"><span class="mi" id="MathJax-Span-2377" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="mo" id="MathJax-Span-2378" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2379" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 4.152em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1004em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.995em;"><span class="mrow" id="MathJax-Span-2380"><span class="mi" id="MathJax-Span-2381" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2382" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-2383" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-2384" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2385" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2386" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-2387" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2388" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1001.8em 4.46em -999.997em); top: -3.327em; left: 50%; margin-left: -0.868em;"><span class="mrow" id="MathJax-Span-2389"><span class="msubsup" id="MathJax-Span-2390"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2391" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2392" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2393"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2394" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2395" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.15em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.152em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.247em; border-left: 0px solid; width: 0px; height: 3.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>ρ</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>σ</mi><mi>x</mi></msub><msub><mi>σ</mi><mi>y</mi></msub></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-209">\rho = \frac{Cov(X, Y)}{\sigma_x\sigma_y}</script></span></p><p>The variance can also be computed as</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-210-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>&#x00B1;</mo><mi>b</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x00B1;</mo><mn>2</mn><mi>a</mi><mi>b</mi><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2396" style="width: 27.769em; display: inline-block;"><span style="display: inline-block; position: relative; width: 22.748em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1022.7em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2397"><span class="mi" id="MathJax-Span-2398" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-2399" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2400" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2401" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2402" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2403" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2404" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mi" id="MathJax-Span-2405" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">b</span><span class="mi" id="MathJax-Span-2406" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2407" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2408" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-2409" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2410" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-2411" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-2412" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-2413" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2414" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2415" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2416" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2417" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2418" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="msubsup" id="MathJax-Span-2419" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2420" style="font-family: STIXGeneral; font-style: italic;">b</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-2421" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-2422" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-2423" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2424" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2425" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2426" style="font-family: STIXGeneral; font-style: italic;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2427" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2428" style="font-family: STIXGeneral; padding-left: 0.259em;">±</span><span class="mn" id="MathJax-Span-2429" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mi" id="MathJax-Span-2430" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-2431" style="font-family: STIXGeneral; font-style: italic;">b</span><span class="mi" id="MathJax-Span-2432" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-2433" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-2434" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-2435" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2436" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2437" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-2438" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">Y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2439" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.441em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>±</mo><mi>b</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>±</mo><mn>2</mn><mi>a</mi><mi>b</mi><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-210">Var(aX \pm bY) = a^2Var(X) + b^2Var(Y) \pm 2abCov(X, Y)</script></span></p><h2 id="distributional-approximations"><a class="anchor hidden-xs" href="#distributional-approximations" title="distributional-approximations"><span class="octicon octicon-link"></span></a>Distributional approximations</h2><h3 id="mean-and-variance-of-binomial-and-poisson-distributions"><a class="anchor hidden-xs" href="#mean-and-variance-of-binomial-and-poisson-distributions" title="mean-and-variance-of-binomial-and-poisson-distributions"><span class="octicon octicon-link"></span></a>Mean and variance of binomial and Poisson distributions</h3><p>Proof of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-211-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi><mo>=</mo><mi>n</mi><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2440" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1002.77em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2441"><span class="mi" id="MathJax-Span-2442" style="font-family: STIXGeneral; font-style: italic;">μ</span><span class="mo" id="MathJax-Span-2443" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2444" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-2445" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>n</mi><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-211">\mu = np</script></span> for binomial</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-212-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>&#x03BC;</mi></mtd><mtd><mi></mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mo>&#x2211;</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>x</mi><mo>&#x00D7;</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mi>x</mi><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>&#x00D7;</mo><msup><mi>p</mi><mi>x</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>&#x00D7;</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>y</mi><mo>&#x2212;</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mi>y</mi><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mi>y</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>&#x00D7;</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>y</mi></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mi>y</mi><mo>&#x2212;</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow></munderover><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2446" style="width: 22.595em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.496em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-6.913em 1018.34em 10.812em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2447"><span class="mtable" id="MathJax-Span-2448" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 18.189em; height: 0px;"><span style="position: absolute; clip: rect(9.121em 1000.46em 26.283em -999.997em); top: -17.877em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -12.139em; right: 0em;"><span class="mtd" id="MathJax-Span-2449"><span class="mrow" id="MathJax-Span-2450"><span class="mi" id="MathJax-Span-2451" style="font-family: STIXGeneral; font-style: italic;">μ</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -10.653em; right: 0em;"><span class="mtd" id="MathJax-Span-2460"><span class="mrow" id="MathJax-Span-2461"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -8.194em; right: 0em;"><span class="mtd" id="MathJax-Span-2473"><span class="mrow" id="MathJax-Span-2474"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -5.018em; right: 0em;"><span class="mtd" id="MathJax-Span-2517"><span class="mrow" id="MathJax-Span-2518"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -1.637em; right: 0em;"><span class="mtd" id="MathJax-Span-2573"><span class="mrow" id="MathJax-Span-2574"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: 1.898em; right: 0em;"><span class="mtd" id="MathJax-Span-2631"><span class="mrow" id="MathJax-Span-2632"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: 4.255em; right: 0em;"><span class="mtd" id="MathJax-Span-2659"><span class="mrow" id="MathJax-Span-2660"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 17.882em;"></span></span><span style="position: absolute; clip: rect(9.275em 1017.68em 26.898em -999.997em); top: -18.286em; left: 0.515em;"><span style="display: inline-block; position: relative; width: 17.677em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1003.23em 4.306em -999.997em); top: -12.139em; left: 0em;"><span class="mtd" id="MathJax-Span-2452"><span class="mrow" id="MathJax-Span-2453"><span class="mi" id="MathJax-Span-2454"></span><span class="mo" id="MathJax-Span-2455" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2456" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2457" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2458" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-2459" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.871em 1004.31em 4.613em -999.997em); top: -10.653em; left: 0em;"><span class="mtd" id="MathJax-Span-2462"><span class="mrow" id="MathJax-Span-2463"><span class="mi" id="MathJax-Span-2464"></span><span class="mo" id="MathJax-Span-2465" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-2466" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="msubsup" id="MathJax-Span-2467" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2468" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mi" id="MathJax-Span-2469" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-2470"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2471" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-2472" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">i</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.257em 1014.86em 5.433em -999.997em); top: -8.194em; left: 0em;"><span class="mtd" id="MathJax-Span-2475"><span class="mrow" id="MathJax-Span-2476"><span class="mi" id="MathJax-Span-2477"></span><span class="mo" id="MathJax-Span-2478" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="munderover" id="MathJax-Span-2479" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2480" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2481"><span class="mrow" id="MathJax-Span-2482"><span class="mi" id="MathJax-Span-2483" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2484" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-2485" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-2486" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-2487" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2488" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mfrac" id="MathJax-Span-2489" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 4.05em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.407em;"><span class="mrow" id="MathJax-Span-2490"><span class="mi" id="MathJax-Span-2491" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2492" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1003.85em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -1.944em;"><span class="mrow" id="MathJax-Span-2493"><span class="mi" id="MathJax-Span-2494" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2495" style="font-family: STIXGeneral;">!</span><span class="mo" id="MathJax-Span-2496" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2497" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2498" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2499" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2500" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2501" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.05em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2502" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-2503" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2504" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mi" id="MathJax-Span-2505" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2506" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-2507" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2508" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2509" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">p</span><span class="msubsup" id="MathJax-Span-2510"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2511" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="texatom" id="MathJax-Span-2512"><span class="mrow" id="MathJax-Span-2513"><span class="mi" id="MathJax-Span-2514" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2515" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2516" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.257em 1017.68em 5.433em -999.997em); top: -5.018em; left: 0em;"><span class="mtd" id="MathJax-Span-2519"><span class="mrow" id="MathJax-Span-2520"><span class="mi" id="MathJax-Span-2521"></span><span class="mo" id="MathJax-Span-2522" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2523" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-2524" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="munderover" id="MathJax-Span-2525" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2526" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2527"><span class="mrow" id="MathJax-Span-2528"><span class="mi" id="MathJax-Span-2529" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2530" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-2531" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-2532" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-2533" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 6.406em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1003.13em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.586em;"><span class="mrow" id="MathJax-Span-2534"><span class="mo" id="MathJax-Span-2535" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2536" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2537" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2538" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2539" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2540" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1006.2em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -3.122em;"><span class="mrow" id="MathJax-Span-2541"><span class="mo" id="MathJax-Span-2542" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2543" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2544" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2545" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2546" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2547" style="font-family: STIXGeneral;">!</span><span class="mo" id="MathJax-Span-2548" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2549" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2550" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2551" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2552" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2553" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1006.41em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 6.406em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2554" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-2555" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.744em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2556" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="texatom" id="MathJax-Span-2557"><span class="mrow" id="MathJax-Span-2558"><span class="mi" id="MathJax-Span-2559" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2560" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2561" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2562" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-2563" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2564" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2565" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">p</span><span class="msubsup" id="MathJax-Span-2566"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2567" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="texatom" id="MathJax-Span-2568"><span class="mrow" id="MathJax-Span-2569"><span class="mi" id="MathJax-Span-2570" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2571" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2572" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.205em 1017.01em 5.689em -999.997em); top: -1.79em; left: 0em;"><span class="mtd" id="MathJax-Span-2575"><span class="mrow" id="MathJax-Span-2576"><span class="mi" id="MathJax-Span-2577"></span><span class="mo" id="MathJax-Span-2578" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2579" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-2580" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="munderover" id="MathJax-Span-2581" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2582" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.408em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2583"><span class="mrow" id="MathJax-Span-2584"><span class="mi" id="MathJax-Span-2585" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-2586" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2587" style="font-size: 70.7%; font-family: STIXGeneral;">0</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1001.13em 4.255em -999.997em); top: -5.172em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2588"><span class="mrow" id="MathJax-Span-2589"><span class="mi" id="MathJax-Span-2590" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2591" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2592" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-2593" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 5.74em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1003.13em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.586em;"><span class="mrow" id="MathJax-Span-2594"><span class="mo" id="MathJax-Span-2595" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2596" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2597" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2598" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2599" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2600" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.53em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -2.815em;"><span class="mrow" id="MathJax-Span-2601"><span class="mi" id="MathJax-Span-2602" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-2603" style="font-family: STIXGeneral;">!</span><span class="mo" id="MathJax-Span-2604" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2605" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2606" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2607" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">y</span><span class="mo" id="MathJax-Span-2608" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2609" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2610" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-2611" style="font-family: STIXGeneral;">!</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1005.74em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.74em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2612" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-2613" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2614" style="font-family: STIXGeneral; font-style: italic;">p</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="texatom" id="MathJax-Span-2615"><span class="mrow" id="MathJax-Span-2616"><span class="mi" id="MathJax-Span-2617" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2618" style="font-family: STIXGeneral;">(</span><span class="mn" id="MathJax-Span-2619" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-2620" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-2621" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">p</span><span class="msubsup" id="MathJax-Span-2622"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2623" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="texatom" id="MathJax-Span-2624"><span class="mrow" id="MathJax-Span-2625"><span class="mi" id="MathJax-Span-2626" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2627" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2628" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-2629" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2630" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span><span style="position: absolute; clip: rect(2.103em 1008.4em 5.484em -999.997em); top: 1.847em; left: 0em;"><span class="mtd" id="MathJax-Span-2633"><span class="mrow" id="MathJax-Span-2634"><span class="mi" id="MathJax-Span-2635"></span><span class="mo" id="MathJax-Span-2636" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2637" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-2638" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="munderover" id="MathJax-Span-2639" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-2640" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2641"><span class="mrow" id="MathJax-Span-2642"><span class="mi" id="MathJax-Span-2643" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2644" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-2645" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1001.13em 4.255em -999.997em); top: -5.172em; left: 0.054em;"><span class="texatom" id="MathJax-Span-2646"><span class="mrow" id="MathJax-Span-2647"><span class="mi" id="MathJax-Span-2648" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2649" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2650" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-2651" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">B</span><span class="mo" id="MathJax-Span-2652" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2653" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2654" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2655" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2656" style="font-family: STIXGeneral;">,</span><span class="mi" id="MathJax-Span-2657" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">p</span><span class="mo" id="MathJax-Span-2658" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 4.05em;"></span></span><span style="position: absolute; clip: rect(3.384em 1002.26em 4.357em -999.997em); top: 4.255em; left: 0em;"><span class="mtd" id="MathJax-Span-2661"><span class="mrow" id="MathJax-Span-2662"><span class="mi" id="MathJax-Span-2663"></span><span class="mo" id="MathJax-Span-2664" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2665" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-2666" style="font-family: STIXGeneral; font-style: italic;">p</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 18.291em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -10.372em; border-left: 0px solid; width: 0px; height: 21.378em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>μ</mi></mtd><mtd><mi></mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>x</mi><mo>×</mo><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mi>x</mi><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>×</mo><msup><mi>p</mi><mi>x</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>×</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>y</mi><mo>−</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow><mrow><mi>y</mi><mo>!</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>y</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo>×</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mi>y</mi></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mi>y</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>x</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>n</mi><mi>p</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-212">\begin{align}\mu &= E(X) \\ &= \sum x_i p_i \\
&= \sum_{x=1}^n x \times \frac{n!}{x!(n-x)!} \times p^x(1-p)^{n-x} \\ &= np\sum_{x=1}^n \frac{(n-1)!}{(x-1)!(n-x)!} \times p^{x-1}(1-p)^{n-x} \\
&= np\sum_{y-0}^{n-1} \frac{(n-1)!}{y!(n-y-1)!} \times p^{y}(1-p)^{n-y-1}\\
&= np\sum_{x=1}^{n-1}B(n-1, p) \\
&= np\end{align}</script></span></p><p>Proof of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-213-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&#x03C3;</mi><mn>2</mn></msup><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mi>p</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2667" style="width: 7.585em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.201em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1006.15em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2668"><span class="msubsup" id="MathJax-Span-2669"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2670" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="mn" id="MathJax-Span-2671" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2672" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2673" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mo" id="MathJax-Span-2674" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-2675" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-2676" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mn" id="MathJax-Span-2677" style="font-family: STIXGeneral; padding-left: 0.259em;">1</span><span class="mo" id="MathJax-Span-2678" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-2679" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-213">\sigma^2 = n(n-1)p</script></span> for binomial</p></div>
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