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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<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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<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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<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 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>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<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>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter contains-active">
|
||
<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>
|
||
<ul class="structural toc-item-list contains-active">
|
||
<li class="toc-item toc-section">
|
||
<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>
|
||
<ul 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>
|
||
</li>
|
||
<li class="toc-item toc-section contains-active">
|
||
<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>
|
||
<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<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>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<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>
|
||
</li>
|
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</ul>
|
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<li class="toc-item toc-chapter">
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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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<ul class="structural toc-item-list">
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<li class="toc-item toc-section">
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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>
|
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</li>
|
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<li class="toc-item toc-section">
|
||
<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>
|
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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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<ul class="structural toc-item-list">
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<li class="toc-item toc-appendix">
|
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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>
|
||
</li>
|
||
<li class="toc-item toc-appendix">
|
||
<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>
|
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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">
|
||
<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>
|
||
</div>
|
||
<article class="example example-like" id="sec-Two-Sample-Tests-3"><h3 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">
|
||
<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*}
|
||
\Var(A_1) \amp = \frac{\sigma^2}{n_1} = \frac{12}{50} = 0.24.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Similarly, let <span class="process-math">\(A_2\)</span> be the random variable representing average height of the experimental group of plants. This sample has size <span class="process-math">\(n_2 = 30\text{,}\)</span> and we measured the average height <span class="process-math">\(\mu_2 = 36.4\text{.}\)</span> The variance of the average height would be:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-3-1-2-8">
|
||
\begin{align*}
|
||
\Var(A_2) \amp = \frac{\sigma^2}{n_2} = \frac{12}{30} = 0.4.
|
||
\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="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*}
|
||
D = A_2 - A_1 = 36.4 - 35 = 1.4
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">The variance would be:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-3-1-3-3">
|
||
\begin{gather*}
|
||
\Var(D) = \Var(A_2 - A_1) = \Var(A_2) + \Var(A_1) = 0.24 + 0.4 = 0.64.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">So, under the null hypothesis, we will treat <span class="process-math">\(D\)</span> as <span class="process-math">\(\Norm(0, 0.64)\text{.}\)</span> We then perform a 1-tailed test:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-3-1-3-6">
|
||
\begin{align*}
|
||
\Pr(D \geq 1.4) \amp \approx \Pr\left(Z \geq \frac{1.4 - 0}{\sqrt{0.64}}\right) \\
|
||
\amp = 1 - \Phi(1.75) \\
|
||
\amp = 1 - 0.9599 \\
|
||
\amp = 0.0401 \lt 0.05,
|
||
\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>
|
||
<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>
|
||
<article class="example example-like" id="sec-Two-Sample-Tests-5"><h3 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>
|
||
<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">
|
||
\begin{gather*}
|
||
\widehat{\theta}_1 = \frac{10}{100} = 0.1.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">If <span class="process-math">\(\theta_2\)</span> is the proportion of left-handed people in the patient population, then the maximum likelihood estimate would be:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-4">
|
||
\begin{gather*}
|
||
\widehat{\theta}_2 = \frac{16}{100} = 0.16.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Write <span class="process-math">\(A_1\)</span> for the proportion of the control group sample which is left-handed and <span class="process-math">\(A_2\)</span> for the proportion of the experimental group sample. Then we can estimate the variances:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-7">
|
||
\begin{align*}
|
||
n_1 \amp = 100 \amp n_2 \amp = 100 \\
|
||
\Var(A_1) \amp = \frac{n_1 \widehat{\theta}_1(1 - \widehat{\theta}_1)}{n_1^2} \amp \Var(A_2) \amp = \frac{n_2 \widehat{\theta}_2(1 - \widehat{\theta}_2)}{n_2^2} \\
|
||
\amp = \frac{(0.1)(0.9)}{100} \amp \amp = \frac{(0.16)(0.84)}{100} \\
|
||
\amp = 0.0009 \amp \amp = 0.001344
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Now, if <span class="process-math">\(D = A_2 - A_1\text{,}\)</span> then:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-9">
|
||
\begin{gather*}
|
||
\Var(D) = \Var(A_2) + \Var(A_1),
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">however, we should pause a moment here. The null hypothesis would be that the two populations have the same statistics. That should include the same variance for an individual measurement. (The variances of <span class="process-math">\(A_1\)</span> and <span class="process-math">\(A_2\)</span> could still be different if, e.g., they were averaging over different sized samples.) So we should abandon our two distinct maximum likelihood estimates <span class="process-math">\(\widehat{\theta}_1, \widehat{\theta}_2\)</span> and instead find one estimate, <span class="process-math">\(\widehat{\theta}_0\text{,}\)</span> estimated from all collected data:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-14">
|
||
\begin{gather*}
|
||
\widehat{\theta}_0 = \frac{10 + 16}{100 + 100} = \frac{26}{200} = 0.13.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Then, we should recalculate the variances of <span class="process-math">\(A_1, A_2\text{:}\)</span>
|
||
</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-16">
|
||
\begin{align*}
|
||
\Var(A_1) \amp = \frac{(0.13)(0.87)}{100} = 0.001131 \\
|
||
\Var(A_2) \amp = \frac{(0.13)(0.87)}{100} = 0.001131
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Finally:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-17">
|
||
\begin{gather*}
|
||
\Var(D) = \Var(A_1) + \Var(A_2) = 0.002262.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Now we can apply a 1-tailed test:</div>
|
||
<div class="displaymath process-math" id="sec-Two-Sample-Tests-5-1-1-18">
|
||
\begin{align*}
|
||
\Pr(D \geq 0.06) \amp \approx \Pr\left( Z \geq \frac{0.06 - 0}{\sqrt{0.002262}}\right) \\
|
||
\amp \approx 1 - \Phi(1.27) \\
|
||
\amp \approx 1 - 0.8962 \\
|
||
\amp = 0.1038 \gt 0.05,
|
||
\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>
|
||
<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">
|
||
<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>
|
||
</div>
|
||
<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>
|
||
<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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