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@@ -9,7 +9,7 @@
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@@ -129,7 +129,13 @@ eBookConfig.enable_chatcodes = false;
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<span class="nav-other-controls"><button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe51c;</span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="sec-Two-Sample-Tests.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cb;</span><span class="name">Prev</span></a><a class="up-button button" href="ch-Hypothesis-Testing.html" title="Up"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5ce;</span><span class="name">Up</span></a><a class="next-button button" href="sec-Chi-Squared.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true">&#xe5cc;</span></a></span>
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<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Introduction.html" class="internal"><span class="codenumber">0.1</span> <span class="title">Introduction</span></a></div></li></ul>
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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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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
</ul>
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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>
<ul 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>
</li>
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<li class="toc-item toc-chapter">
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<li class="toc-item toc-chapter">
<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>
<ul class="structural toc-item-list">
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<li class="toc-item toc-section"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Confidence Intervals</span></a></div></li>
<li class="toc-item toc-section">
<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.1</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">4.1</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">4.2</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">4.2</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>
<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></li>
<li class="toc-item toc-section"><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></li>
<li class="toc-item toc-section active"><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></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></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>
<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>
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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>
<ul class="structural toc-item-list"><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>
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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 active"><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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<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>
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<li class="toc-item toc-chapter">
<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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<li class="toc-item toc-section"><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></li>
<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></li>
<li class="toc-item toc-section">
<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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<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>
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@@ -255,7 +294,34 @@ eBookConfig.enable_chatcodes = false;
</h2>
<div class="para" id="sec-Power-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Power-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Section 5.3: Power of a Test"><a href="#sec-Power" title="Copy heading and permalink for Section 5.3: Power of a Test" aria-label="Copy heading and permalink for Section 5.3: Power of a Test">🔗</a></div></section></div>
<section class="exercises" id="exercises-Power"><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-Power-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-Power-1-1-1">Suppose we find a coin and wonder whether it’s fair. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-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 5.3.1"><a href="#exercises-Power-1" title="Copy heading and permalink for Exercise 5.3.1" aria-label="Copy heading and permalink for Exercise 5.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-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-Power-2-1-1">Suppose we have a coin which we suspect comes up heads more often than a fair coin would. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-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 5.3.2"><a href="#exercises-Power-2" title="Copy heading and permalink for Exercise 5.3.2" aria-label="Copy heading and permalink for Exercise 5.3.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-Power-3-1"><div class="para" id="exercises-Power-3-1-1">Suppose we find a six-sided die and wonder whether it’s fair. As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. The expected number of 1’s is 50/3, with a variance of 125/9.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<article class="task exercise-like" id="exercises-Power-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-Power-3-2-1-1">Using a normal approximation, what is the smallest number of 1’s greater than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 5.3.3.a"><a href="#exercises-Power-3-2" title="Copy heading and permalink for Task 5.3.3.a" aria-label="Copy heading and permalink for Task 5.3.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-Power-3-3-1-1">Using a normal approximation, what is the greatest number of 1’s less than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 5.3.3.b"><a href="#exercises-Power-3-3" title="Copy heading and permalink for Task 5.3.3.b" aria-label="Copy heading and permalink for Task 5.3.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-Power-3-4-1-1">Suppose that this die is weighted so that it rolls a 1 with probability 0.2. What would be the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 5.3.3.c"><a href="#exercises-Power-3-4" title="Copy heading and permalink for Task 5.3.3.c" aria-label="Copy heading and permalink for Task 5.3.3.c">🔗</a></div></article><article class="task exercise-like" id="exercises-Power-3-5"><h5 class="heading"><span class="codenumber">(d)</span></h5>
<div class="para" id="exercises-Power-3-5-1-1">Suppose we roll the die 100 times and see 23 1’s. Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-3-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 5.3.3.d"><a href="#exercises-Power-3-5" title="Copy heading and permalink for Task 5.3.3.d" aria-label="Copy heading and permalink for Task 5.3.3.d">🔗</a></div></article><div class="autopermalink" data-description="Exercise 5.3.3"><a href="#exercises-Power-3" title="Copy heading and permalink for Exercise 5.3.3" aria-label="Copy heading and permalink for Exercise 5.3.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Power-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Power-4-1-1">Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<span class="process-math">\(^2\text{.}\)</span> We suspect that growing this plant in a greenhouse will increase its height. We take the average height of a sample of 50 plants grown in a greenhouse. What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the <span class="process-math">\(p = 0.05\)</span> significance level? If the plants, when grown in a greenhouse, would truly have an average height of 41 in, what is the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Power-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 5.3.4"><a href="#exercises-Power-4" title="Copy heading and permalink for Exercise 5.3.4" aria-label="Copy heading and permalink for Exercise 5.3.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 5.3"><a href="#exercises-Power" title="Copy heading and permalink for Exercises 5.3" aria-label="Copy heading and permalink for Exercises 5.3">🔗</a></div></section><div class="autopermalink" data-description="Section 5.3: Power of a Test"><a href="#sec-Power" title="Copy heading and permalink for Section 5.3: Power of a Test" aria-label="Copy heading and permalink for Section 5.3: Power of a Test">🔗</a></div></section></div>
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