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@@ -9,7 +9,7 @@
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<!--******************************************-->
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<!--* Authored with PreTeXt *-->
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<!--* pretextbook.org *-->
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<!--* Theme: boulder *-->
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<!--* Palette: *-->
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<!--******************************************-->
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<head xmlns:og="http://ogp.me/ns#" xmlns:book="https://ogp.me/ns/book#">
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||||
@@ -101,15 +101,15 @@ eBookConfig.allow_pairs = false;
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eBookConfig.enableScratchAC = false;
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eBookConfig.build_info = "";
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eBookConfig.python3 = null;
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||||
eBookConfig.runestone_version = '7.11.5';
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||||
eBookConfig.runestone_version = '7.11.8';
|
||||
eBookConfig.jobehost = '';
|
||||
eBookConfig.proxyuri_runs = '';
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||||
eBookConfig.proxyuri_files = '';
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||||
eBookConfig.enable_chatcodes = false;
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||||
</script>
|
||||
<!--*** Runestone Services ***-->
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||||
<script src="_static/prefix-runtime.42217a82c75796c8.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.7e141ad8414abf4a.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.f4ab138da65f4203.css">
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<script src="_static/prefix-runtime.991c135ccf182bf5.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.1c010553c35c544c.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
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<link rel="stylesheet" type="text/css" href="_static/prefix-runestone.677a75a9120449ff.css">
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<script src="_static/pretext/js/lti_iframe_resizer.js"></script><script src="https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min.js" integrity="sha512-4xUl/d6D6THrAnXAwGajXkoWaeMNwEKK4iNfq5DotEbLPAfk6FSxSP3ydNxqDgCw1c/0Z1Jg6L8h2j+++9BZmg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script><script src="lunr-pretext-search-index.js" async=""></script><script src="_static/pretext/js/pretext_search.js"></script><script src="_static/pretext/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
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||||
</script>
|
||||
</head>
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||||
@@ -129,7 +129,13 @@ eBookConfig.enable_chatcodes = false;
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||||
<ol id="searchresults" class="searchresults"></ol>
|
||||
</div>
|
||||
</div>
|
||||
<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"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="sec-Power.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></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"></span><span class="name">Up</span></a><a class="next-button button" href="ch-Linear-Regression.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
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<span class="nav-other-controls"><button id="embed-button" class="embed-button button" title="Embed this page"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Embed</span></button><div class="embed-popup hidden" id="embed-popup">
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<p>Copy the code below to embed this page in your own website or LMS page.</p>
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||||
<div class="embed-code-container">
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<textarea class="embed-code-textbox" id="embed-code-textbox" readonly aria-label="textbox"><iframe src="https://example.com/embed" width="100%" height="1000"></iframe></textarea><button class="copy-embed-button button" id="copy-embed-button" title="Copy embed code"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Copy</span></button>
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||||
</div>
|
||||
</div>
|
||||
<button id="light-dark-button" class="light-dark-button button" title="Dark Mode"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Dark Mode</span></button></span><span class="treebuttons"><a class="previous-button button" href="sec-Power.html" title="Previous"><span class="icon material-symbols-outlined" aria-hidden="true"></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"></span><span class="name">Up</span></a><a class="next-button button" href="ch-Linear-Regression.html" title="Next"><span class="name">Next</span><span class="icon material-symbols-outlined" aria-hidden="true"></span></a></span>
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</div></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
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\newcommand{\Z}{\mathbb Z}
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||||
\newcommand{\Q}{\mathbb Q}
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@@ -144,6 +150,7 @@ eBookConfig.enable_chatcodes = false;
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\Cov}{Cov}
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\newcommand{\lt}{<}
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\newcommand{\gt}{>}
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||||
\newcommand{\amp}{&}
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@@ -151,9 +158,13 @@ eBookConfig.enable_chatcodes = false;
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\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
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\)</span></div>
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||||
<div class="ptx-page">
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||||
<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2 focused" data-preexpanded-levels="0" data-max-levels="2"><ul class="structural toc-item-list contains-active">
|
||||
<div id="ptx-sidebar" class="ptx-sidebar"><nav id="ptx-toc" class="ptx-toc depth2"><ul class="structural toc-item-list contains-active">
|
||||
<li class="toc-item toc-frontmatter"><div class="toc-title-box"><a href="frontmatter.html" class="internal"><span class="title">Front Matter</span></a></div></li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
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||||
<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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||||
</li>
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||||
<li class="toc-item toc-chapter">
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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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||||
<ul class="structural toc-item-list">
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||||
<li class="toc-item toc-section">
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||||
@@ -201,6 +212,10 @@ eBookConfig.enable_chatcodes = false;
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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>
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||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-section">
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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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||||
<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>
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||||
</li>
|
||||
</ul>
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||||
</li>
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||||
<li class="toc-item toc-chapter">
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||||
@@ -228,24 +243,48 @@ eBookConfig.enable_chatcodes = false;
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||||
<li class="toc-item toc-chapter">
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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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||||
<ul class="structural toc-item-list">
|
||||
<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></li>
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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>
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||||
<li class="toc-item toc-section">
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||||
<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>
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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">4.1</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">
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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>
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||||
<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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||||
</li>
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||||
</ul>
|
||||
</li>
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||||
<li class="toc-item toc-chapter contains-active">
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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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||||
<ul class="structural toc-item-list contains-active">
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||||
<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>
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||||
<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>
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||||
<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></li>
|
||||
<li class="toc-item toc-section active"><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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||||
<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>
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||||
</li>
|
||||
<li class="toc-item toc-section">
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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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||||
<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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||||
</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>
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||||
<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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||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
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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>
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||||
<ul class="structural toc-item-list active"><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>
|
||||
</ul>
|
||||
</li>
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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>
|
||||
<ul class="structural toc-item-list">
|
||||
<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>
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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></li>
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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>
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||||
<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">
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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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||||
<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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||||
</li>
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||||
</ul>
|
||||
</li>
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||||
<li class="toc-item toc-backmatter"><div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div></li>
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||||
@@ -255,7 +294,61 @@ eBookConfig.enable_chatcodes = false;
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</h2>
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<div class="para" id="sec-Chi-Squared-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Chi-Squared-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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||||
</div>
|
||||
<div class="autopermalink" data-description="Section 5.4: \chi^2 Test"><a href="#sec-Chi-Squared" title="Copy heading and permalink for Section 5.4: \chi^2 Test" aria-label="Copy heading and permalink for Section 5.4: \chi^2 Test">🔗</a></div></section></div>
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||||
<section class="exercises" id="exercises-Chi-Squared"><h3 class="heading hide-type">
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<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>
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</h3>
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<article class="exercise exercise-like" id="exercises-Chi-Squared-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="exercises-Chi-Squared-1-1-1">A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. They survey their customers about their favorite drinks. Is the data below consistent with the null hypothesis that each type of drink will be equally preferred?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Chi-Squared-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> <figure class="table table-like" id="exercises-Chi-Squared-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.4.1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 5.4.1: "><a href="#exercises-Chi-Squared-1-1-2" title="Copy heading and permalink for Table 5.4.1: " aria-label="Copy heading and permalink for Table 5.4.1: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
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||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines">drink type</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">water</th>
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<th scope="col" class="c m b1 r0 l0 t0 lines">soda</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">tea</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">coffee</th>
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<th scope="col" class="c m b1 r0 l0 t0 lines">energy drinks</th>
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</tr>
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<tr>
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<td class="c m b0 r0 l0 t0 lines">favorite</td>
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||||
<td class="c m b0 r0 l0 t0 lines">28</td>
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||||
<td class="c m b0 r0 l0 t0 lines">17</td>
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||||
<td class="c m b0 r0 l0 t0 lines">15</td>
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||||
<td class="c m b0 r0 l0 t0 lines">26</td>
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||||
<td class="c m b0 r0 l0 t0 lines">14</td>
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</tr>
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||||
</table></div></figure><div class="autopermalink" data-description="Exercise 5.4.1"><a href="#exercises-Chi-Squared-1" title="Copy heading and permalink for Exercise 5.4.1" aria-label="Copy heading and permalink for Exercise 5.4.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Chi-Squared-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="introduction" id="exercises-Chi-Squared-2-1">
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||||
<div class="para" id="exercises-Chi-Squared-2-1-1">A particular drug is administered in 100 independent trials. In each trial, the drug is administered to four people, and we count how many respond to the drug. The table below shows how many trials have each different count of people who respond to the drug.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Chi-Squared-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>
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||||
<figure class="table table-like" id="exercises-Chi-Squared-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">5.4.2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 5.4.2: "><a href="#exercises-Chi-Squared-2-1-2" title="Copy heading and permalink for Table 5.4.2: " aria-label="Copy heading and permalink for Table 5.4.2: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr class="header-horizontal">
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines"># who respond to drug</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">0</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">1</th>
|
||||
<th scope="col" class="c m b1 r0 l0 t0 lines">2</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">3</th>
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||||
<th scope="col" class="c m b1 r0 l0 t0 lines">4</th>
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||||
</tr>
|
||||
<tr>
|
||||
<td class="c m b0 r0 l0 t0 lines"># of trials</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">11</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">31</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">34</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">21</td>
|
||||
</tr>
|
||||
</table></div></figure>
|
||||
</div>
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||||
<article class="task exercise-like" id="exercises-Chi-Squared-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
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<div class="para" id="exercises-Chi-Squared-2-2-1-1">Is the data consistent with a binomial distribution with parameter <span class="process-math">\(p = 0.7\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Chi-Squared-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 5.4.2.a"><a href="#exercises-Chi-Squared-2-2" title="Copy heading and permalink for Task 5.4.2.a" aria-label="Copy heading and permalink for Task 5.4.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Chi-Squared-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
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<div class="para" id="exercises-Chi-Squared-2-3-1-1">What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation for the probability <span class="process-math">\(p\)</span> that a person will respond to the drug?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Chi-Squared-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 5.4.2.b"><a href="#exercises-Chi-Squared-2-3" title="Copy heading and permalink for Task 5.4.2.b" aria-label="Copy heading and permalink for Task 5.4.2.b">🔗</a></div></article><article class="task exercise-like" id="exercises-Chi-Squared-2-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
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<div class="para" id="exercises-Chi-Squared-2-4-1-1">Is the data consistent with a binomial distribution with the MLE value of <span class="process-math">\(p\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Chi-Squared-2-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 5.4.2.c"><a href="#exercises-Chi-Squared-2-4" title="Copy heading and permalink for Task 5.4.2.c" aria-label="Copy heading and permalink for Task 5.4.2.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 5.4.2"><a href="#exercises-Chi-Squared-2" title="Copy heading and permalink for Exercise 5.4.2" aria-label="Copy heading and permalink for Exercise 5.4.2">🔗</a></div></article><div class="autopermalink" data-description="Exercises 5.4"><a href="#exercises-Chi-Squared" title="Copy heading and permalink for Exercises 5.4" aria-label="Copy heading and permalink for Exercises 5.4">🔗</a></div></section><div class="autopermalink" data-description="Section 5.4: \chi^2 Test"><a href="#sec-Chi-Squared" title="Copy heading and permalink for Section 5.4: \chi^2 Test" aria-label="Copy heading and permalink for Section 5.4: \chi^2 Test">🔗</a></div></section></div>
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