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2026-02-10 07:59:07 -05:00
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@@ -62,7 +62,7 @@ window.MathJax = {
"[pretext]/mathjaxknowl3.js"
],
"paths": {
"pretext": "_static/pretext/js/lib"
"pretext": "_static/pretext/js"
}
},
"startup": {
@@ -79,7 +79,7 @@ window.MathJax = {
<meta property="og:type" content="book">
<meta property="book:title" content="Math 1044 Notes">
<meta property="book:author" content="Andy Eisenberg">
<script src="_static/pretext/js/lib/jquery.min.js"></script><script src="_static/pretext/js/lib/jquery.sticky.js"></script><script src="_static/pretext/js/lib/jquery.espy.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></script><script src="_static/pretext/js/user_preferences.js"></script><!--** eBookCongig is necessary to configure interactive **-->
<script src="_static/pretext/js/jquery.min.js"></script><script src="_static/pretext/js/pretext.js"></script><script src="_static/pretext/js/pretext_add_on.js?x=1"></script><!--** eBookCongig is necessary to configure interactive **-->
<!--** Runestone components to run locally in reader's browser **-->
<!--** No external communication: **-->
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@@ -101,16 +101,16 @@ eBookConfig.allow_pairs = false;
eBookConfig.enableScratchAC = false;
eBookConfig.build_info = "";
eBookConfig.python3 = null;
eBookConfig.runestone_version = '7.11.11';
eBookConfig.runestone_version = '7.11.13';
eBookConfig.jobehost = '';
eBookConfig.proxyuri_runs = '';
eBookConfig.proxyuri_files = '';
eBookConfig.enable_chatcodes = false;
</script>
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<script src="_static/prefix-runtime.fc0e080e84d54281.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.css">
<script src="_static/prefix-runtime.0d5f811896ddeac1.bundle.js"></script><script src="_static/prefix-723.3e6434f80549315a.bundle.js"></script><script src="_static/prefix-runestone.e2abc79debce15fd.bundle.js"></script><link rel="stylesheet" type="text/css" href="_static/prefix-723.3bccd435914aa0ff.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)';
<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/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script>sagecellEvalName='Evaluate (Sage)';
</script>
</head>
<body class="pretext book ignore-math">
@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
<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">&#xe14d;</span><span class="name">Copy</span></button>
</div>
</div>
<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="ch-Confidence-Intervals.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-Confidence-Intervals.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-Confidence-Intervals.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></nav><div id="latex-macros" class="hidden-content process-math" style="display:none"><span class="process-math">\(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
@@ -151,6 +151,9 @@ eBookConfig.enable_chatcodes = false;
\DeclareMathOperator{\E}{E}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\newcommand{\L}{\mathcal{L}}
\newcommand{\est}{\widehat}
\newcommand{\lt}{&lt;}
\newcommand{\gt}{&gt;}
\newcommand{\amp}{&amp;}
@@ -243,13 +246,17 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-chapter contains-active">
<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 contains-active">
<li class="toc-item toc-section">
<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>
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Likelihood.html#sec-Likelihood-20" class="internal"><span class="codenumber">4.1</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-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 active"><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>
<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 active"><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">4.2</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>
<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>
@@ -290,7 +297,7 @@ eBookConfig.enable_chatcodes = false;
<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>
</ul></nav></div>
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-CLT"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Central Limit Theorem</span>
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Central Limit Theorem</span>
</h2>
<div class="para" id="sec-CLT-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
@@ -300,24 +307,24 @@ eBookConfig.enable_chatcodes = false;
<article class="exercise exercise-like" id="exercises-CLT-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-CLT-1-1-1">Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. Let <span class="process-math">\(S\)</span> be the number of heads. Estimate the probability that <span class="process-math">\(34 \leq S \leq 44\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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 4.1.1"><a href="#exercises-CLT-1" title="Copy heading and permalink for Exercise 4.1.1" aria-label="Copy heading and permalink for Exercise 4.1.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="autopermalink" data-description="Exercise 4.2.1"><a href="#exercises-CLT-1" title="Copy heading and permalink for Exercise 4.2.1" aria-label="Copy heading and permalink for Exercise 4.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-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-CLT-2-1-1">Suppose a fair die is rolled 100 times, and let <span class="process-math">\(m\)</span> be the average value of the rolls. Estimate the probability that <span class="process-math">\(3.45 \leq m \leq 3.55\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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 4.1.2"><a href="#exercises-CLT-2" title="Copy heading and permalink for Exercise 4.1.2" aria-label="Copy heading and permalink for Exercise 4.1.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="autopermalink" data-description="Exercise 4.2.2"><a href="#exercises-CLT-2" title="Copy heading and permalink for Exercise 4.2.2" aria-label="Copy heading and permalink for Exercise 4.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-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-CLT-3-1"><div class="para" id="exercises-CLT-3-1-1">The heights of men in the US have a mean of 69 in and a variance of about 9 in<span class="process-math">\(^2\text{,}\)</span> and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in<span class="process-math">\(^2\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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-CLT-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-CLT-3-2-1-1">Suppose the heights of 30 men are sampled, and a sample mean <span class="process-math">\(m\)</span> is taken. Find the expected value and variance of <span class="process-math">\(m\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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 4.1.3.a"><a href="#exercises-CLT-3-2" title="Copy heading and permalink for Task 4.1.3.a" aria-label="Copy heading and permalink for Task 4.1.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="autopermalink" data-description="Task 4.2.3.a"><a href="#exercises-CLT-3-2" title="Copy heading and permalink for Task 4.2.3.a" aria-label="Copy heading and permalink for Task 4.2.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-CLT-3-3-1-1">Estimate the probability that <span class="process-math">\(m \geq 69.5\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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 4.1.3.b"><a href="#exercises-CLT-3-3" title="Copy heading and permalink for Task 4.1.3.b" aria-label="Copy heading and permalink for Task 4.1.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="autopermalink" data-description="Task 4.2.3.b"><a href="#exercises-CLT-3-3" title="Copy heading and permalink for Task 4.2.3.b" aria-label="Copy heading and permalink for Task 4.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-CLT-3-4-1-1">What if the sampled group was women?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-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 4.1.3.c"><a href="#exercises-CLT-3-4" title="Copy heading and permalink for Task 4.1.3.c" aria-label="Copy heading and permalink for Task 4.1.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 4.1.3"><a href="#exercises-CLT-3" title="Copy heading and permalink for Exercise 4.1.3" aria-label="Copy heading and permalink for Exercise 4.1.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.1"><a href="#exercises-CLT" title="Copy heading and permalink for Exercises 4.1" aria-label="Copy heading and permalink for Exercises 4.1">🔗</a></div></section><div class="autopermalink" data-description="Section 4.1: Central Limit Theorem"><a href="#sec-CLT" title="Copy heading and permalink for Section 4.1: Central Limit Theorem" aria-label="Copy heading and permalink for Section 4.1: Central Limit Theorem">🔗</a></div></section></div>
<div class="autopermalink" data-description="Task 4.2.3.c"><a href="#exercises-CLT-3-4" title="Copy heading and permalink for Task 4.2.3.c" aria-label="Copy heading and permalink for Task 4.2.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 4.2.3"><a href="#exercises-CLT-3" title="Copy heading and permalink for Exercise 4.2.3" aria-label="Copy heading and permalink for Exercise 4.2.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.2"><a href="#exercises-CLT" title="Copy heading and permalink for Exercises 4.2" aria-label="Copy heading and permalink for Exercises 4.2">🔗</a></div></section><div class="autopermalink" data-description="Section 4.2: Central Limit Theorem"><a href="#sec-CLT" title="Copy heading and permalink for Section 4.2: Central Limit Theorem" aria-label="Copy heading and permalink for Section 4.2: Central Limit Theorem">🔗</a></div></section></div>
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