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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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@@ -189,16 +161,17 @@ eBookConfig.enable_chatcodes = false;
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@@ -208,7 +181,7 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
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@@ -218,16 +191,16 @@ eBookConfig.enable_chatcodes = false;
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<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<ul id="ptx-toc-group-sec-Joint-Distributions" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<ul id="ptx-toc-group-sec-Expected-Value" class="structural toc-item-list">
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
@@ -236,90 +209,96 @@ eBookConfig.enable_chatcodes = false;
</li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Covariance" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Covariance.html#exercises-Covariance" class="internal"><span class="codenumber">3.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Confidence-Intervals-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Likelihood.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Likelihood</span></a></div>
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<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Central Limit Theorem</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-CLT" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
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<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Hypothesis-Testing-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
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<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
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<ul id="ptx-toc-group-sec-One-Sample-Tests" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
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<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Chi-Squared" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
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<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Correlation" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
<ul id="ptx-toc-group-sec-Linear-Regression" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
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<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
<ul id="ptx-toc-group-backmatter-2" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
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<div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="codenumber">B</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values Table</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
<ul id="ptx-toc-group-backmatter-3" class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Chi-squared-table.html" class="internal"><span class="codenumber">B.1</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Critical Values</span></a></div></li></ul>
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<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-4.html" class="internal"><span class="title">Colophon</span></a></div></li>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Two-Sample-Tests"><h2 class="heading hide-type">
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
<section class="section" id="sec-Two-Sample-Tests">
<h1 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Two Sample Tests</span>
</h2>
<div class="para" id="sec-Two-Sample-Tests-2">In the previous section, we compared data collected from a sample against a known or supposed baseline. But the statistics for that baseline would also generally be approximated by collecting data from a sample. We’ll now consider hypothesis tests that directly involve two samples of data, one called a <dfn class="terminology">control group</dfn> and the other an <dfn class="terminology">experimental group</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</h1>
<div class="para" id="sec-Two-Sample-Tests-2">In the previous section, we compared data collected from a sample against a known or supposed baseline. But the statistics for that baseline would also generally be approximated by collecting data from a sample. We’ll now consider hypothesis tests that directly involve two samples of data, one called a <dfn class="terminology">control group</dfn> and the other an <dfn class="terminology">experimental group</dfn>.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-Two-Sample-Tests-3"><h3 class="heading">
<article class="example example-like" id="sec-Two-Sample-Tests-3"><h2 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para" id="sec-Two-Sample-Tests-3-1-1">Suppose a sample of 50 plants have a mean height of 35 cm. A separate sample of 30 plants are taken from another plot which has been using a new fertilizer intended to increase plant growth, and they have a mean height of 36.4 cm. If the height of a plant has a variance of <span class="process-math">\(\sigma^2 = 12 \text{cm}^2\text{,}\)</span> do we have strong enough evidence to reject the null hypothesis that the new fertilizer has no effect on plant height?<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="sec-Two-Sample-Tests-3-1-2">
</h2>
<div class="para" id="sec-Two-Sample-Tests-3-1-1">Suppose a sample of 50 plants have a mean height of 35 cm. A separate sample of 30 plants are taken from another plot which has been using a new fertilizer intended to increase plant growth, and they have a mean height of 36.4 cm. If the height of a plant has a variance of <span class="process-math">\(\sigma^2 = 12 \text{cm}^2\text{,}\)</span> do we have strong enough evidence to reject the null hypothesis that the new fertilizer has no effect on plant height?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="sec-Two-Sample-Tests-3-1-2">
<div class="para">Let <span class="process-math">\(A_1\)</span> be the random variable representing average height of the control group of plants. This sample has size <span class="process-math">\(n_1 = 50\text{,}\)</span> and we measured the average height <span class="process-math">\(\mu_1 = 35\text{.}\)</span> The variance of the average height would be:</div>
<div class="displaymath process-math" id="sec-Two-Sample-Tests-3-1-2-4">
\begin{align*}
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\end{align*}
</div>
<div class="para">Let <span class="process-math">\(D = A_2 - A_1\)</span> be the difference between the average heights.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="sec-Two-Sample-Tests-3-1-3">
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="sec-Two-Sample-Tests-3-1-3">
<div class="para">According to the null hypothesis, the two plots of plants should behave the same way, so <span class="process-math">\(\E(D) = 0\text{.}\)</span> We measured the value:</div>
<div class="displaymath process-math" id="sec-Two-Sample-Tests-3-1-3-2">
\begin{gather*}
@@ -357,13 +337,15 @@ D = A_2 - A_1 = 36.4 - 35 = 1.4
\end{align*}
</div>
<div class="para">so we can reject the null hypothesis.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-3-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 5.2.1"><a href="#sec-Two-Sample-Tests-3" title="Copy heading and permalink for Example 5.2.1" aria-label="Copy heading and permalink for Example 5.2.1">🔗</a></div></article><div class="para" id="sec-Two-Sample-Tests-4">However, we don’t know <span class="process-math">\(\sigma^2\text{,}\)</span> so we should replace it with the sample variance.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" aria-hidden="true" data-description="Example 5.2.1"><a tabindex="-1" href="#sec-Two-Sample-Tests-3" title="Copy heading and permalink for Example 5.2.1" aria-label="Copy heading and permalink for Example 5.2.1">🔗</a></div>
</article>
<div class="para" id="sec-Two-Sample-Tests-4">However, we don’t know <span class="process-math">\(\sigma^2\text{,}\)</span> so we should replace it with the sample variance.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-Two-Sample-Tests-5"><h3 class="heading">
<article class="example example-like" id="sec-Two-Sample-Tests-5"><h2 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
</h2>
<div class="para logical" id="sec-Two-Sample-Tests-5-1-1">
<div class="para">Suppose a sample of 100 people from the general population contains 10 left-handed people, and a sample of 100 patients from a carpal tunnel syndrome clinic contains 16 left-handed people. If <span class="process-math">\(\theta_1\)</span> is the proportion of left-handed people in the general population, then the maximum likelihood estimation based on our control group sample would be:</div>
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-2">
@@ -422,18 +404,27 @@ n_1 \amp = 100 \amp n_2 \amp = 100 \\
\end{align*}
</div>
<div class="para">so we do not reject the null hypothesis.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Two-Sample-Tests-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Two-Sample-Tests-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 5.2.2"><a href="#sec-Two-Sample-Tests-5" title="Copy heading and permalink for Example 5.2.2" aria-label="Copy heading and permalink for Example 5.2.2">🔗</a></div></article><section class="exercises" id="exercises-Two-Sample-Tests"><h3 class="heading hide-type">
<div class="autopermalink" aria-hidden="true" data-description="Example 5.2.2"><a tabindex="-1" href="#sec-Two-Sample-Tests-5" title="Copy heading and permalink for Example 5.2.2" aria-label="Copy heading and permalink for Example 5.2.2">🔗</a></div>
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<section class="exercises" id="exercises-Two-Sample-Tests">
<h2 class="heading hide-type">
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercises-Two-Sample-Tests-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Two-Sample-Tests-1-1-1">A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety. A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams<span class="process-math">\(^2\)</span> (for an individual apple). An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams<span class="process-math">\(^2\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Two-Sample-Tests-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</h2>
<article class="exercise exercise-like" id="exercises-Two-Sample-Tests-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exercises-Two-Sample-Tests-1-1-1">A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety. A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams<span class="process-math">\(^2\)</span> (for an individual apple). An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams<span class="process-math">\(^2\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Two-Sample-Tests-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 5.2.1"><a href="#exercises-Two-Sample-Tests-1" title="Copy heading and permalink for Exercise 5.2.1" aria-label="Copy heading and permalink for Exercise 5.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Two-Sample-Tests-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Two-Sample-Tests-2-1-1">We have an established factory which produces coins that are close to fair. We’re opening up a second factory, and we’d like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. We pick one sample coin from each factory, and flip each sample coin 100 times. The coin from the established factory flips 52 heads in 100 flips. The coin from the new factory flips 62 heads in 100 flips. Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Two-Sample-Tests-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="autopermalink" aria-hidden="true" data-description="Exercise 5.2.1"><a tabindex="-1" href="#exercises-Two-Sample-Tests-1" title="Copy heading and permalink for Exercise 5.2.1" aria-label="Copy heading and permalink for Exercise 5.2.1">🔗</a></div>
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<article class="exercise exercise-like" id="exercises-Two-Sample-Tests-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exercises-Two-Sample-Tests-2-1-1">We have an established factory which produces coins that are close to fair. We’re opening up a second factory, and we’d like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. We pick one sample coin from each factory, and flip each sample coin 100 times. The coin from the established factory flips 52 heads in 100 flips. The coin from the new factory flips 62 heads in 100 flips. Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level?<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Two-Sample-Tests-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" aria-hidden="true" data-description="Exercise 5.2.2"><a tabindex="-1" href="#exercises-Two-Sample-Tests-2" title="Copy heading and permalink for Exercise 5.2.2" aria-label="Copy heading and permalink for Exercise 5.2.2">🔗</a></div>
</article><div class="autopermalink" aria-hidden="true" data-description="Exercises 5.2"><a tabindex="-1" href="#exercises-Two-Sample-Tests" title="Copy heading and permalink for Exercises 5.2" aria-label="Copy heading and permalink for Exercises 5.2">🔗</a></div>
</section><div class="autopermalink" aria-hidden="true" data-description="Section 5.2: Two Sample Tests"><a tabindex="-1" href="#sec-Two-Sample-Tests" title="Copy heading and permalink for Section 5.2: Two Sample Tests" aria-label="Copy heading and permalink for Section 5.2: Two Sample Tests">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 5.2.2"><a href="#exercises-Two-Sample-Tests-2" title="Copy heading and permalink for Exercise 5.2.2" aria-label="Copy heading and permalink for Exercise 5.2.2">🔗</a></div></article><div class="autopermalink" data-description="Exercises 5.2"><a href="#exercises-Two-Sample-Tests" title="Copy heading and permalink for Exercises 5.2" aria-label="Copy heading and permalink for Exercises 5.2">🔗</a></div></section><div class="autopermalink" data-description="Section 5.2: Two Sample Tests"><a href="#sec-Two-Sample-Tests" title="Copy heading and permalink for Section 5.2: Two Sample Tests" aria-label="Copy heading and permalink for Section 5.2: Two Sample Tests">🔗</a></div></section></div>
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