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@@ -19,67 +19,21 @@
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<meta property="book:title" content="Math 1044 Notes">
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<meta property="book:author" content="Andy Eisenberg">
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<body class="pretext book ignore-math">
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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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<p class="byline">Andy Eisenberg</p>
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<a class="assistive" href="#ptx-content">Skip to main content</a><header id="ptx-masthead" class="ptx-masthead" role="banner"><div class="ptx-banner"><div class="title-container">
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<div class="heading ptx-banner-heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></div>
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<p class="byline">Andy Eisenberg (testing)</p>
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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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<legend>Assistive technologies</legend>
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<label for="ptx-readability-accessible-permalinks"><input type="checkbox" id="ptx-readability-accessible-permalinks"><span>🔗</span><span>Expose permalinks</span></label>
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<div class="ptx-readability-options-group"><button type="button" class="ptx-readability-reset-button button" id="ptx-readability-reset-button"><span class="icon material-symbols-outlined" aria-hidden="true"></span><span class="name">Reset All</span></button></div></dialog></span><span class="treebuttons"><a class="previous-button button" href="sec-Likelihood.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-Confidence-Intervals.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="sec-Confidence-Intervals.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">\(\newcommand{\N}{\mathbb N}
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@@ -162,26 +133,27 @@ eBookConfig.enable_chatcodes = false;
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\)</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"><ul class="structural toc-item-list contains-active">
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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>
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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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||||
<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 id="ptx-toc-group-ch-Probability" 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-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set Theory</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-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</span> <span class="title">Exercises</span></a></div></li></ul>
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<ul id="ptx-toc-group-sec-Set-Theory" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Set-Theory.html#exercises-Set-Theory" class="internal"><span class="codenumber">1.1</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-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</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-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<ul id="ptx-toc-group-sec-Probability" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Probability.html#exercises-Probability" class="internal"><span class="codenumber">1.2</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-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<ul class="structural toc-item-list">
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<ul id="ptx-toc-group-sec-Conditional-Probability" class="structural toc-item-list">
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
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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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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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@@ -189,16 +161,17 @@ eBookConfig.enable_chatcodes = false;
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|
||||
<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
|
||||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
<ul id="ptx-toc-group-sec-Independent-Events" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Independent-Events.html#exercises-Independent-Events" class="internal"><span class="codenumber">1.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-ch-Random-Variables" class="structural toc-item-list">
|
||||
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Random-Variables-2.html" class="internal"><span class="title">Introduction</span></a></div></li>
|
||||
<li class="toc-item toc-section">
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-sec-Discrete-RVs" class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
|
||||
@@ -208,7 +181,7 @@ eBookConfig.enable_chatcodes = false;
|
||||
</li>
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-sec-Continuous-RVs" class="structural toc-item-list">
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
|
||||
<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>
|
||||
<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
|
||||
@@ -218,16 +191,16 @@ eBookConfig.enable_chatcodes = false;
|
||||
</li>
|
||||
<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>
|
||||
<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>
|
||||
<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>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-chapter">
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-ch-Expected-Value" class="structural toc-item-list">
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||||
<li class="toc-item toc-section">
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<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>
|
||||
@@ -236,88 +209,93 @@ 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>
|
||||
<ul id="ptx-toc-group-sec-Variance" 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>
|
||||
</li>
|
||||
<li class="toc-item toc-section">
|
||||
<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>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<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">
|
||||
<ul class="structural toc-item-list contains-active" id="ptx-toc-group-ch-Confidence-Intervals">
|
||||
<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>
|
||||
<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>
|
||||
<ul id="ptx-toc-group-sec-Likelihood" 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">4.2</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>
|
||||
<ul class="structural toc-item-list active" id="ptx-toc-group-sec-CLT"><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>
|
||||
<ul id="ptx-toc-group-sec-Confidence-Intervals" 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">
|
||||
<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">
|
||||
<ul id="ptx-toc-group-ch-Hypothesis-Testing" class="structural toc-item-list">
|
||||
<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>
|
||||
<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>
|
||||
<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>
|
||||
</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>
|
||||
<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>
|
||||
<ul id="ptx-toc-group-sec-Two-Sample-Tests" 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>
|
||||
</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>
|
||||
<ul id="ptx-toc-group-sec-Power" 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>
|
||||
<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>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<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>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-ch-Linear-Regression" class="structural toc-item-list">
|
||||
<li class="toc-item toc-introduction"><div class="toc-title-box"><a href="ch-Linear-Regression-2.html" class="internal"><span class="title">Introduction</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>
|
||||
<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>
|
||||
</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>
|
||||
<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>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toc-item toc-backmatter">
|
||||
<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
|
||||
<ul class="structural toc-item-list">
|
||||
<ul id="ptx-toc-group-backmatter" class="structural toc-item-list">
|
||||
<li class="toc-item toc-appendix">
|
||||
<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>
|
||||
</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>
|
||||
<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>
|
||||
</li>
|
||||
<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>
|
||||
</ul>
|
||||
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|
||||
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|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-CLT"><h2 class="heading hide-type">
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
|
||||
<section class="section" id="sec-CLT">
|
||||
<h1 class="heading hide-type">
|
||||
<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">Although we defined the normal distribution earlier (<a href="sec-Continuous-RVs.html#def-normal-distribution" class="xref" data-knowl="./knowl/xref/def-normal-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.2.8">Definition 2.2.8</a>), we haven’t done much in the way of probability calculations with it. The sense in which this distribution is "normal" is that it turns out other distributions, when added together or averaged, tend to be well-approximated by a normal distribution.<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>
|
||||
</h1>
|
||||
<div class="para" id="sec-CLT-2">Although we defined the normal distribution earlier (<a href="sec-Continuous-RVs.html#def-normal-distribution" class="xref" data-knowl="./knowl/xref/def-normal-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.2.8">Definition 2.2.8</a>), we haven’t done much in the way of probability calculations with it. The sense in which this distribution is "normal" is that it turns out other distributions, when added together or averaged, tend to be well-approximated by a normal distribution.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="theorem theorem-like" id="thm-CLT"><h3 class="heading">
|
||||
<article class="theorem theorem-like" id="thm-CLT"><h2 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="thm-CLT-1-1">
|
||||
<div class="para">Suppose <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">iid</dfn>) random variables with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Let:</div>
|
||||
<div class="displaymath process-math" id="thm-CLT-1-1-6">
|
||||
@@ -334,26 +312,34 @@ S_n \approx \Norm(n \mu, n \sigma^2) \\
|
||||
A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-CLT-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="#thm-CLT-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="Theorem 4.2.1"><a href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div></article><article class="remark remark-like" id="sec-CLT-4"><h3 class="heading">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Theorem 4.2.1"><a tabindex="-1" href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div>
|
||||
</article>
|
||||
<article class="remark remark-like" id="sec-CLT-4"><h2 class="heading">
|
||||
<span class="type">Remark</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="sec-CLT-4-1">
|
||||
<div class="para">How large is "sufficiently large"? We’ll make use of two common rules of thumb:</div>
|
||||
<ol class="decimal" id="sec-CLT-4-1-1"><li id="sec-CLT-4-1-1-1">
|
||||
<div class="para" id="sec-CLT-4-1-1-1-1">For general distributions, we’d like <span class="process-math">\(n \geq 30\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<ol class="decimal" id="sec-CLT-4-1-1">
|
||||
<li id="sec-CLT-4-1-1-1">
|
||||
<div class="para" id="sec-CLT-4-1-1-1-1">For general distributions, we’d like <span class="process-math">\(n \geq 30\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-4-1-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#sec-CLT-4-1-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li></ol>
|
||||
<ol class="decimal" id="sec-CLT-4-1-2"><li id="sec-CLT-4-1-2-1">
|
||||
<div class="para" id="sec-CLT-4-1-2-1-1">For a binomial distribution specifically, we’d like <span class="process-math">\(n\)</span> large enough that we have at least five heads and five tails.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1-2-1-1" 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="Item 1"><a tabindex="-1" href="#sec-CLT-4-1-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<ol class="decimal" id="sec-CLT-4-1-2">
|
||||
<li id="sec-CLT-4-1-2-1">
|
||||
<div class="para" id="sec-CLT-4-1-2-1-1">For a binomial distribution specifically, we’d like <span class="process-math">\(n\)</span> large enough that we have at least five heads and five tails.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Item 1"><a href="#sec-CLT-4-1-2-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li></ol>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1" 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="Item 1"><a tabindex="-1" href="#sec-CLT-4-1-2-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
|
||||
</li>
|
||||
</ol>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-4-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Remark 4.2.2"><a href="#sec-CLT-4" title="Copy heading and permalink for Remark 4.2.2" aria-label="Copy heading and permalink for Remark 4.2.2">🔗</a></div></article><div class="para logical" id="sec-CLT-5">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Remark 4.2.2"><a tabindex="-1" href="#sec-CLT-4" title="Copy heading and permalink for Remark 4.2.2" aria-label="Copy heading and permalink for Remark 4.2.2">🔗</a></div>
|
||||
</article>
|
||||
<div class="para logical" id="sec-CLT-5">
|
||||
<div class="para">When a normal distribution gets translated and/or scaled, the result is another normal distribution. It will often be useful to <em class="emphasis">standardize</em> a normal distribution <span class="process-math">\(X = \Norm(\mu, \sigma^2)\)</span> as follows:</div>
|
||||
<div class="displaymath process-math" id="sec-CLT-5-3">
|
||||
\begin{gather*}
|
||||
@@ -362,11 +348,11 @@ Z = \frac{X - \mu}{\sigma} = \Norm(0, 1).
|
||||
</div>
|
||||
<div class="para">Given an <span class="process-math">\(X\)</span>-value <span class="process-math">\(a\text{,}\)</span> the corresponding value of <span class="process-math">\(Z\)</span> is called a <dfn class="terminology"><span class="process-math">\(z\)</span>-score</dfn>, and is sometimes denoted <span class="process-math">\(a^*\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-5" 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-CLT-5" 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-CLT-6"><h3 class="heading">
|
||||
<article class="example example-like" id="sec-CLT-6"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="sec-CLT-6-1-1">
|
||||
<div class="para">Suppose we flip a fair coin 100 times and let <span class="process-math">\(S_{100}\)</span> be the number of heads. In order to calculate, for example, <span class="process-math">\(\Pr(40 \leq S_{100} \leq 60)\)</span> using the binomial distribution probability formula, we would need to calculate:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/thm-CLT.html" id="sec-CLT-6-1-1-3">
|
||||
@@ -386,9 +372,11 @@ Z \amp = \frac{S_{100} - 50}{\sqrt{25}} \\
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">We now reach a potential impass.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-6-1-1" 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-CLT-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 4.2.3"><a href="#sec-CLT-6" title="Copy heading and permalink for Example 4.2.3" aria-label="Copy heading and permalink for Example 4.2.3">🔗</a></div></article><div class="para logical" id="sec-CLT-7">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 4.2.3"><a tabindex="-1" href="#sec-CLT-6" title="Copy heading and permalink for Example 4.2.3" aria-label="Copy heading and permalink for Example 4.2.3">🔗</a></div>
|
||||
</article>
|
||||
<div class="para logical" id="sec-CLT-7">
|
||||
<div class="para">The random variable <span class="process-math">\(Z\)</span> is the <em class="emphasis">standard normal distribution</em>, with pdf:</div>
|
||||
<div class="displaymath process-math" id="sec-CLT-7-3">
|
||||
\begin{gather*}
|
||||
@@ -414,15 +402,15 @@ Z \amp = \frac{S_{100} - 50}{\sqrt{25}} \\
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">Unfortunately, we don’t have a nice formula for <span class="process-math">\(\Phi(x)\text{.}\)</span> In fact, there is provably no "nice" formula that could be written for <span class="process-math">\(\Phi(x)\)</span> (where "nice" has some technical mathematical meaning that we won’t discuss here).</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-7" 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-CLT-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="sec-CLT-8">The true power of <a href="sec-CLT.html#thm-CLT" class="xref" data-knowl="./knowl/xref/thm-CLT.html" data-reveal-label="Reveal" data-close-label="Close" title="Theorem 4.2.1">Theorem 4.2.1</a> comes to the rescue: it doesn’t matter what type of distribution the individual <span class="process-math">\(X_i\)</span>’s have, and it doesn’t matter what type of distributions <span class="process-math">\(S_n\)</span> and <span class="process-math">\(A_n\)</span> have. The random variable <span class="process-math">\(Z\)</span> is <span class="process-math">\(\Norm(0, 1)\text{.}\)</span> Since it’s the same particular distribution all the time, and since we’re only aiming at an approximation anyway, we can calculate many estimations of probabilities like <span class="process-math">\(\Pr(a \leq Z \leq b)\)</span> ahead of time and just save the results to look up later.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="para" id="sec-CLT-8">The true power of <a href="sec-CLT.html#thm-CLT" class="xref" data-knowl="./knowl/xref/thm-CLT.html" data-reveal-label="Reveal" data-close-label="Close" title="Theorem 4.2.1">Theorem 4.2.1</a> comes to the rescue: it doesn’t matter what type of distribution the individual <span class="process-math">\(X_i\)</span>’s have, and it doesn’t matter what type of distributions <span class="process-math">\(S_n\)</span> and <span class="process-math">\(A_n\)</span> have. The random variable <span class="process-math">\(Z\)</span> is <span class="process-math">\(\Norm(0, 1)\text{.}\)</span> Since it’s the same particular distribution all the time, and since we’re only aiming at an approximation anyway, we can calculate many estimations of probabilities like <span class="process-math">\(\Pr(a \leq Z \leq b)\)</span> ahead of time and just save the results to look up later.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="sec-CLT-9">A table of precalculated <span class="process-math">\(\Phi(z)\)</span> values can be found in <a href="app-Phi-table.html" class="internal" title="Section A.1: \Phi(z) Table of Values">Section A.1</a>.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-9" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="para" id="sec-CLT-9">A table of precalculated <span class="process-math">\(\Phi(z)\)</span> values can be found in <a href="app-Phi-table.html" class="internal" title="Section A.1: \Phi(z) Table of Values">Section A.1</a>.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-CLT-9" 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-CLT-10"><h3 class="heading">
|
||||
<article class="example example-like" id="sec-CLT-10"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="sec-CLT-10-1-1">
|
||||
<div class="para">Continuing the previous example, and having looked up the values <span class="process-math">\(\Phi(2) \approx 0.9772\)</span> and <span class="process-math">\(\Phi(-2) \approx 0.0228\text{,}\)</span> we complete our estimation:</div>
|
||||
<div class="displaymath process-math" id="sec-CLT-10-1-1-3">
|
||||
@@ -434,9 +422,11 @@ Z \amp = \frac{S_{100} - 50}{\sqrt{25}} \\
|
||||
\amp = 0.9544.
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-10-1-1" 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-CLT-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 4.2.4"><a href="#sec-CLT-10" title="Copy heading and permalink for Example 4.2.4" aria-label="Copy heading and permalink for Example 4.2.4">🔗</a></div></article><div class="para logical" id="sec-CLT-11">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 4.2.4"><a tabindex="-1" href="#sec-CLT-10" title="Copy heading and permalink for Example 4.2.4" aria-label="Copy heading and permalink for Example 4.2.4">🔗</a></div>
|
||||
</article>
|
||||
<div class="para logical" id="sec-CLT-11">
|
||||
<div class="para">The true value is approximately <span class="process-math">\(0.965\text{,}\)</span> so we underestimated the probability a bit. It turns out this will consistently be the case when using a normal approximation to a discrete distribution, such as the binomial distribution. The range of values <span class="process-math">\(40 \leq S_{100} \leq 60\)</span> is a list of <span class="process-math">\(21\)</span> discrete values. However, the continuous interval <span class="process-math">\([40, 60]\)</span> has a length of only 20. So calculating the <span class="process-math">\(z\)</span>-scores from the values <span class="process-math">\(40\)</span> and <span class="process-math">\(60\)</span> has the effect of shortening the interval length by 1. To mitigate this, we can make a <dfn class="terminology">continuity correction</dfn> by extending the values out by <span class="process-math">\(0.5\)</span> in each direction:</div>
|
||||
<div class="displaymath process-math" id="sec-CLT-11-10">
|
||||
\begin{align*}
|
||||
@@ -448,60 +438,91 @@ Z \amp = \frac{S_{100} - 50}{\sqrt{25}} \\
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="para">This is a much more accurate estimate.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-11" 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-CLT-11" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<section class="exercises" id="exercises-CLT"><h3 class="heading hide-type">
|
||||
<section class="exercises" id="exercises-CLT">
|
||||
<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-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>
|
||||
</h2>
|
||||
<article class="exercise exercise-like" id="exercises-CLT-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-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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="solutions"><details id="exercises-CLT-1-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="solutions">
|
||||
<details id="exercises-CLT-1-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-CLT-1-2-1">
|
||||
<span class="process-math">\(\Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\(\Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294.\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-CLT-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.2.1.1"><a href="#exercises-CLT-1-2" title="Copy heading and permalink for Answer 4.2.1.1" aria-label="Copy heading and permalink for Answer 4.2.1.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<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>
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<div class="autopermalink" aria-hidden="true" data-description="Answer 4.2.1.1"><a tabindex="-1" href="#exercises-CLT-1-2" title="Copy heading and permalink for Answer 4.2.1.1" aria-label="Copy heading and permalink for Answer 4.2.1.1">🔗</a></div>
|
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</div></details>
|
||||
</div>
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||||
<div class="solutions"><details id="exercises-CLT-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
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<div class="autopermalink" aria-hidden="true" data-description="Exercise 4.2.1"><a tabindex="-1" 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>
|
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</article>
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<article class="exercise exercise-like" id="exercises-CLT-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-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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="solutions">
|
||||
<details id="exercises-CLT-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
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<div class="para" id="exercises-CLT-2-2-1">
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<span class="process-math">\(\Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\(\Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510.\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-CLT-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.2.2.1"><a href="#exercises-CLT-2-2" title="Copy heading and permalink for Answer 4.2.2.1" aria-label="Copy heading and permalink for Answer 4.2.2.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<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 class="autopermalink" aria-hidden="true" data-description="Answer 4.2.2.1"><a tabindex="-1" href="#exercises-CLT-2-2" title="Copy heading and permalink for Answer 4.2.2.1" aria-label="Copy heading and permalink for Answer 4.2.2.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-CLT-3-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Exercise 4.2.2"><a tabindex="-1" 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"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="solutions">
|
||||
<details id="exercises-CLT-3-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-CLT-3-2-2-1">
|
||||
<span class="process-math">\(\E(m) = 69\)</span> and <span class="process-math">\(\Var(m) = 0.3\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\(\E(m) = 69\)</span> and <span class="process-math">\(\Var(m) = 0.3\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-CLT-3-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.2.3.a.1"><a href="#exercises-CLT-3-2-2" title="Copy heading and permalink for Answer 4.2.3.a.1" aria-label="Copy heading and permalink for Answer 4.2.3.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<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 class="autopermalink" aria-hidden="true" data-description="Answer 4.2.3.a.1"><a tabindex="-1" href="#exercises-CLT-3-2-2" title="Copy heading and permalink for Answer 4.2.3.a.1" aria-label="Copy heading and permalink for Answer 4.2.3.a.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-CLT-3-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Task 4.2.3.a"><a tabindex="-1" 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"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<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" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="solutions">
|
||||
<details id="exercises-CLT-3-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-CLT-3-3-2-1">
|
||||
<span class="process-math">\(\Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\(\Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-CLT-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.2.3.b.1"><a href="#exercises-CLT-3-3-2" title="Copy heading and permalink for Answer 4.2.3.b.1" aria-label="Copy heading and permalink for Answer 4.2.3.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<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">If the sampled group was women, estimate the probability that <span class="process-math">\(m \geq 64\text{.}\)</span><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 class="autopermalink" aria-hidden="true" data-description="Answer 4.2.3.b.1"><a tabindex="-1" href="#exercises-CLT-3-3-2" title="Copy heading and permalink for Answer 4.2.3.b.1" aria-label="Copy heading and permalink for Answer 4.2.3.b.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-CLT-3-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Task 4.2.3.b"><a tabindex="-1" 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"><h4 class="heading"><span class="codenumber">(c)</span></h4>
|
||||
<div class="para" id="exercises-CLT-3-4-1-1">If the sampled group was women, estimate the probability that <span class="process-math">\(m \geq 64\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" 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="solutions">
|
||||
<details id="exercises-CLT-3-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary>
|
||||
<div class="answer solution-like knowl__content">
|
||||
<div class="para" id="exercises-CLT-3-4-2-1">
|
||||
<span class="process-math">\(\E(m) = 63.5\)</span> and <span class="process-math">\(\Var(m) \approx 0.208\text{.}\)</span> <span class="process-math">\(\Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\(\E(m) = 63.5\)</span> and <span class="process-math">\(\Var(m) \approx 0.208\text{.}\)</span> <span class="process-math">\(\Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-CLT-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Answer 4.2.3.c.1"><a tabindex="-1" href="#exercises-CLT-3-4-2" title="Copy heading and permalink for Answer 4.2.3.c.1" aria-label="Copy heading and permalink for Answer 4.2.3.c.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Task 4.2.3.c"><a tabindex="-1" 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" aria-hidden="true" data-description="Exercise 4.2.3"><a tabindex="-1" 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" aria-hidden="true" data-description="Exercises 4.2"><a tabindex="-1" 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" aria-hidden="true" data-description="Section 4.2: Central Limit Theorem"><a tabindex="-1" 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>
|
||||
<div class="autopermalink" data-description="Answer 4.2.3.c.1"><a href="#exercises-CLT-3-4-2" title="Copy heading and permalink for Answer 4.2.3.c.1" aria-label="Copy heading and permalink for Answer 4.2.3.c.1">🔗</a></div>
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||||
</div></details></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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