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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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</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 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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<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 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 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 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">
|
||||
<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>
|
||||
@@ -236,84 +209,89 @@ 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">
|
||||
<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"><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 id="ptx-toc-group-sec-CLT" class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||||
</li>
|
||||
<li class="toc-item toc-section contains-active">
|
||||
<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 active"><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 class="structural toc-item-list active" id="ptx-toc-group-sec-Confidence-Intervals"><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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|
||||
</ul></nav></div>
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Confidence-Intervals"><h2 class="heading hide-type">
|
||||
<main class="ptx-main"><div id="ptx-content" class="ptx-content">
|
||||
<section class="section" id="sec-Confidence-Intervals">
|
||||
<h1 class="heading hide-type">
|
||||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Confidence Intervals</span>
|
||||
</h2>
|
||||
<div class="para" id="sec-Confidence-Intervals-2">When estimating the value of some parameter, the maximum likelihood estimation provides the single estimate with highest possible likelihood. But we might prefer to have a range of values, with some quantitative measure of how certain we are that the true parameter value could be in the range.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</h1>
|
||||
<div class="para" id="sec-Confidence-Intervals-2">When estimating the value of some parameter, the maximum likelihood estimation provides the single estimate with highest possible likelihood. But we might prefer to have a range of values, with some quantitative measure of how certain we are that the true parameter value could be in the range.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Confidence-Intervals-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical" id="sec-Confidence-Intervals-3">
|
||||
<div class="para">Suppose some measurement has a mean <span class="process-math">\(\mu\)</span> and variance <span class="process-math">\(\sigma^2\)</span> across the entire population. Independent measurements <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> and calculate the sample mean</div>
|
||||
@@ -324,13 +302,13 @@ A_n = \frac{X_1 + \dotsb + X_n}{n}.
|
||||
</div>
|
||||
<div class="para">Then, according to <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>, <span class="process-math">\(A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)\)</span> as long as we take a sufficient number of measurements. However, we don’t have access to the true values of <span class="process-math">\(\mu, \sigma^2\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-3" 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-Confidence-Intervals-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para" id="sec-Confidence-Intervals-4">We can make an estimate <span class="process-math">\(s^2\)</span> of <span class="process-math">\(\sigma^2\)</span> from the collected data, so we know how spread out the normal distribution should be.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<div class="para" id="sec-Confidence-Intervals-4">We can make an estimate <span class="process-math">\(s^2\)</span> of <span class="process-math">\(\sigma^2\)</span> from the collected data, so we know how spread out the normal distribution should be.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Confidence-Intervals-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<article class="definition definition-like" id="def-sample-variance"><h3 class="heading">
|
||||
<article class="definition definition-like" id="def-sample-variance"><h2 class="heading">
|
||||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="def-sample-variance-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> be independent samples taken from a population. Let <span class="process-math">\(A_n = \frac{X_1 + \dotsb + X_n}{n}\)</span> be the sample mean. The <dfn class="terminology">sample variance</dfn> is:</div>
|
||||
<div class="displaymath process-math" id="def-sample-variance-1-1-4">
|
||||
@@ -345,9 +323,11 @@ s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right
|
||||
\end{gather*}
|
||||
</div>
|
||||
<div class="para">is called the <dfn class="terminology">standard error of the mean</dfn>.</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#def-sample-variance-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="#def-sample-variance-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="Definition 4.3.1"><a href="#def-sample-variance" title="Copy heading and permalink for Definition 4.3.1" aria-label="Copy heading and permalink for Definition 4.3.1">🔗</a></div></article><div class="para" id="sec-Confidence-Intervals-6">Note that <span class="process-math">\(s^2\)</span> is an estimate of the variance of each individual measurement <span class="process-math">\(X_i\)</span> taken from the population. Therefore, <span class="process-math">\(\frac{s^2}{n}\)</span> is an estimate of the variance of the mean <span class="process-math">\(A_n\text{.}\)</span> Hence "standard error of the mean" is an estimate of the standard deviation of the mean.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-6" 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="Definition 4.3.1"><a tabindex="-1" href="#def-sample-variance" title="Copy heading and permalink for Definition 4.3.1" aria-label="Copy heading and permalink for Definition 4.3.1">🔗</a></div>
|
||||
</article>
|
||||
<div class="para" id="sec-Confidence-Intervals-6">Note that <span class="process-math">\(s^2\)</span> is an estimate of the variance of each individual measurement <span class="process-math">\(X_i\)</span> taken from the population. Therefore, <span class="process-math">\(\frac{s^2}{n}\)</span> is an estimate of the variance of the mean <span class="process-math">\(A_n\text{.}\)</span> Hence "standard error of the mean" is an estimate of the standard deviation of the mean.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Confidence-Intervals-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="para logical" id="sec-Confidence-Intervals-7">
|
||||
<div class="para">Next, we want to pick a distance <span class="process-math">\(c\)</span> so that our measured sample mean <span class="process-math">\(A_n\)</span> is within <span class="process-math">\(c\)</span> of the true population mean <span class="process-math">\(\mu\)</span> with a high probability. We can pick some value <span class="process-math">\(\alpha \in (0, 1)\text{,}\)</span> and let <span class="process-math">\(c_{\alpha} = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right)\text{.}\)</span> Then the interval:</div>
|
||||
@@ -358,11 +338,11 @@ s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right
|
||||
</div>
|
||||
<div class="para">is called a <dfn class="terminology">confidence interval</dfn> with confidence level <span class="process-math">\(1 - \alpha\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-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-Confidence-Intervals-7" 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-CI"><h3 class="heading">
|
||||
<article class="theorem theorem-like" id="thm-CI"><h2 class="heading">
|
||||
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="thm-CI-1-1">
|
||||
<div class="para">Let <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> be independent samples taken from a population. Suppose <span class="process-math">\(A_n\)</span> is the sample mean and <span class="process-math">\(s^2\)</span> the sample variance. Then the 95% confidence limits are:</div>
|
||||
<div class="displaymath process-math" id="thm-CI-1-1-4">
|
||||
@@ -371,13 +351,16 @@ s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right
|
||||
\mu_{h} \amp = A_n + 1.96 \left(\frac{s}{\sqrt{n}}\right)
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#thm-CI-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-CI-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 4.3.2"><a href="#thm-CI" title="Copy heading and permalink for Theorem 4.3.2" aria-label="Copy heading and permalink for Theorem 4.3.2">🔗</a></div></article><article class="example example-like" id="sec-Confidence-Intervals-9"><h3 class="heading">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Theorem 4.3.2"><a tabindex="-1" href="#thm-CI" title="Copy heading and permalink for Theorem 4.3.2" aria-label="Copy heading and permalink for Theorem 4.3.2">🔗</a></div>
|
||||
</article>
|
||||
<article class="example example-like" id="sec-Confidence-Intervals-9"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
<div class="para" id="sec-Confidence-Intervals-9-1-1">Suppose measurements are taken of the heights of several plants in a garden plot. The measurements are given below. (Note: for the sake of a managable calculation, only five measurements are given. However, for <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> to apply, the number of measurements would need to be greater.)<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="sec-Confidence-Intervals-9-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Plant Height Measurements<div class="autopermalink" data-description="Table 4.3.4: Plant Height Measurements"><a href="#sec-Confidence-Intervals-9-1-2" title="Copy heading and permalink for Table 4.3.4: Plant Height Measurements" aria-label="Copy heading and permalink for Table 4.3.4: Plant Height Measurements">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
</h2>
|
||||
<div class="para" id="sec-Confidence-Intervals-9-1-1">Suppose measurements are taken of the heights of several plants in a garden plot. The measurements are given below. (Note: for the sake of a managable calculation, only five measurements are given. However, for <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> to apply, the number of measurements would need to be greater.)<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#sec-Confidence-Intervals-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="sec-Confidence-Intervals-9-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Plant Height Measurements<div class="autopermalink" aria-hidden="true" data-description="Table 4.3.4: Plant Height Measurements"><a tabindex="-1" href="#sec-Confidence-Intervals-9-1-2" title="Copy heading and permalink for Table 4.3.4: Plant Height Measurements" aria-label="Copy heading and permalink for Table 4.3.4: Plant Height Measurements">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<tr>
|
||||
<td class="c m b1 r0 l0 t0 lines">Plant <span class="process-math">\(i\)</span>
|
||||
</td>
|
||||
@@ -403,7 +386,9 @@ s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5\)</span></td>
|
||||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(10.4\)</span></td>
|
||||
</tr>
|
||||
</table></div></figure> <div class="para logical" id="sec-Confidence-Intervals-9-1-3">
|
||||
</table></div>
|
||||
</figure>
|
||||
<div class="para logical" id="sec-Confidence-Intervals-9-1-3">
|
||||
<div class="para">Then the sample mean is:</div>
|
||||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-1">
|
||||
\begin{align*}
|
||||
@@ -435,9 +420,11 @@ s^2 = \frac{\left(\sum H_i\right)^2 - 5 (A_5)^2}{5 - 1} = \frac{481.1 - 5(9.8)^2
|
||||
\mu_{h} \amp = 9.8 + 1.96(0.212) \approx 10.22
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-9-1-3" 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-Confidence-Intervals-9-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 4.3.3"><a href="#sec-Confidence-Intervals-9" title="Copy heading and permalink for Example 4.3.3" aria-label="Copy heading and permalink for Example 4.3.3">🔗</a></div></article><div class="para logical" id="sec-Confidence-Intervals-10">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 4.3.3"><a tabindex="-1" href="#sec-Confidence-Intervals-9" title="Copy heading and permalink for Example 4.3.3" aria-label="Copy heading and permalink for Example 4.3.3">🔗</a></div>
|
||||
</article>
|
||||
<div class="para logical" id="sec-Confidence-Intervals-10">
|
||||
<div class="para">If the individual measurements <span class="process-math">\(X_1, \dotsc, X_n\)</span> are indicator random variables, then the mean <span class="process-math">\(A_n\)</span> represents the proportion which take on the value 1. For example, if the <span class="process-math">\(X_i\)</span>’s indicate heads on coin flips, then <span class="process-math">\(A_n\)</span> represents the proportion which come up heads. In this setting, we have a nice shortcut formula (<a href="sec-Variance.html#example-binomial-variance" class="xref" data-knowl="./knowl/xref/example-binomial-variance.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 3.2.7">Example 3.2.7</a>) for the variance of <span class="process-math">\(S_n = X_1 + \dotsc + X_n\text{.}\)</span> We can make use of this formula to get the variance of <span class="process-math">\(A_n\)</span> without going through the sample variance calculation above:</div>
|
||||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-binomial-variance.html" id="sec-Confidence-Intervals-10-8">
|
||||
\begin{align*}
|
||||
@@ -450,11 +437,11 @@ s^2 = \frac{\left(\sum H_i\right)^2 - 5 (A_5)^2}{5 - 1} = \frac{481.1 - 5(9.8)^2
|
||||
</div>
|
||||
<div class="para">Since we won’t have access to the true value of <span class="process-math">\(p\text{,}\)</span> we can use the maximum likelihood estimate <span class="process-math">\(\widehat{p}\text{.}\)</span>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-10" 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-Confidence-Intervals-10" 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-Confidence-Intervals-11"><h3 class="heading">
|
||||
<article class="example example-like" id="sec-Confidence-Intervals-11"><h2 class="heading">
|
||||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||||
</h3>
|
||||
</h2>
|
||||
<div class="para logical" id="sec-Confidence-Intervals-11-1-1">
|
||||
<div class="para">Suppose in a sample of 100 people, 40 have a particular trait. The maximum likelihood estimate for the probability <span class="process-math">\(p\)</span> of a random person having the trait is <span class="process-math">\(\widehat{p} = \frac{40}{100} = 0.4\text{.}\)</span> So the standard error will be:</div>
|
||||
<div class="displaymath process-math" id="sec-Confidence-Intervals-11-1-1-3">
|
||||
@@ -469,44 +456,66 @@ p_{\ell} \amp = 0.4 - 1.96(0.049) \approx 0.304 \\
|
||||
p_{h} \amp = 0.4 + 1.96(0.049) \approx 0.496
|
||||
\end{align*}
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Paragraph"><a href="#sec-Confidence-Intervals-11-1-1" 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-Confidence-Intervals-11-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Example 4.3.5"><a href="#sec-Confidence-Intervals-11" title="Copy heading and permalink for Example 4.3.5" aria-label="Copy heading and permalink for Example 4.3.5">🔗</a></div></article><section class="exercises" id="exercises-Confidence-Intervals"><h3 class="heading hide-type">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Example 4.3.5"><a tabindex="-1" href="#sec-Confidence-Intervals-11" title="Copy heading and permalink for Example 4.3.5" aria-label="Copy heading and permalink for Example 4.3.5">🔗</a></div>
|
||||
</article>
|
||||
<section class="exercises" id="exercises-Confidence-Intervals">
|
||||
<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-Confidence-Intervals-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-Confidence-Intervals-1-1-1">Suppose we flip a coin 100 times and count 60 heads. Let <span class="process-math">\(p\)</span> be the (unknown) probability that the coin comes up heads on a flip. Give an approximate 95% confidence interval for the value of <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-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-Confidence-Intervals-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-Confidence-Intervals-1-1-1">Suppose we flip a coin 100 times and count 60 heads. Let <span class="process-math">\(p\)</span> be the (unknown) probability that the coin comes up heads on a flip. Give an approximate 95% confidence interval for the value of <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-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-Confidence-Intervals-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-Confidence-Intervals-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-Confidence-Intervals-1-2-1">
|
||||
<span class="process-math">\([0.504, 0.696]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\([0.504, 0.696]\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-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 4.3.1.1"><a href="#exercises-Confidence-Intervals-1-2" title="Copy heading and permalink for Answer 4.3.1.1" aria-label="Copy heading and permalink for Answer 4.3.1.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 4.3.1"><a href="#exercises-Confidence-Intervals-1" title="Copy heading and permalink for Exercise 4.3.1" aria-label="Copy heading and permalink for Exercise 4.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Confidence-Intervals-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-Confidence-Intervals-2-1"><div class="para" id="exercises-Confidence-Intervals-2-1-1">Suppose in a sample of 100 people, 12 are left-handed.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-2-1-1" 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-Confidence-Intervals-2-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
|
||||
<div class="para" id="exercises-Confidence-Intervals-2-2-1-1">Give a 95% confidence interval for the proportion <span class="process-math">\(p\)</span> of left-handed people.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-2-2-1-1" 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 4.3.1.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-1-2" title="Copy heading and permalink for Answer 4.3.1.1" aria-label="Copy heading and permalink for Answer 4.3.1.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-Confidence-Intervals-2-2-2" 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 4.3.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-1" title="Copy heading and permalink for Exercise 4.3.1" aria-label="Copy heading and permalink for Exercise 4.3.1">🔗</a></div>
|
||||
</article>
|
||||
<article class="exercise exercise-like" id="exercises-Confidence-Intervals-2"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="introduction" id="exercises-Confidence-Intervals-2-1">
|
||||
<div class="para" id="exercises-Confidence-Intervals-2-1-1">Suppose in a sample of 100 people, 12 are left-handed.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-1-1" 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-Confidence-Intervals-2-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
|
||||
<div class="para" id="exercises-Confidence-Intervals-2-2-1-1">Give a 95% confidence interval for the proportion <span class="process-math">\(p\)</span> of left-handed people.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions">
|
||||
<details id="exercises-Confidence-Intervals-2-2-2" 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-Confidence-Intervals-2-2-2-1">
|
||||
<span class="process-math">\([0.056, 0.184]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\([0.056, 0.184]\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.3.2.a.1"><a href="#exercises-Confidence-Intervals-2-2-2" title="Copy heading and permalink for Answer 4.3.2.a.1" aria-label="Copy heading and permalink for Answer 4.3.2.a.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 4.3.2.a"><a href="#exercises-Confidence-Intervals-2-2" title="Copy heading and permalink for Task 4.3.2.a" aria-label="Copy heading and permalink for Task 4.3.2.a">🔗</a></div></article><article class="task exercise-like" id="exercises-Confidence-Intervals-2-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
|
||||
<div class="para" id="exercises-Confidence-Intervals-2-3-1-1">Give a 90% confidence interval.<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-2-3-1-1" 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 4.3.2.a.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-2-2" title="Copy heading and permalink for Answer 4.3.2.a.1" aria-label="Copy heading and permalink for Answer 4.3.2.a.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="solutions"><details id="exercises-Confidence-Intervals-2-3-2" 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 4.3.2.a"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-2" title="Copy heading and permalink for Task 4.3.2.a" aria-label="Copy heading and permalink for Task 4.3.2.a">🔗</a></div>
|
||||
</article>
|
||||
<article class="task exercise-like" id="exercises-Confidence-Intervals-2-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
|
||||
<div class="para" id="exercises-Confidence-Intervals-2-3-1-1">Give a 90% confidence interval.<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="solutions">
|
||||
<details id="exercises-Confidence-Intervals-2-3-2" 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-Confidence-Intervals-2-3-2-1">
|
||||
<span class="process-math">\([0.066, 0.174]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
<span class="process-math">\([0.066, 0.174]\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.3.2.b.1"><a href="#exercises-Confidence-Intervals-2-3-2" title="Copy heading and permalink for Answer 4.3.2.b.1" aria-label="Copy heading and permalink for Answer 4.3.2.b.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Task 4.3.2.b"><a href="#exercises-Confidence-Intervals-2-3" title="Copy heading and permalink for Task 4.3.2.b" aria-label="Copy heading and permalink for Task 4.3.2.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 4.3.2"><a href="#exercises-Confidence-Intervals-2" title="Copy heading and permalink for Exercise 4.3.2" aria-label="Copy heading and permalink for Exercise 4.3.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Confidence-Intervals-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Confidence-Intervals-3-1-1">The weights of five mice are measured and recorded below. Give a 95% confidence interval for the sample mean weight of mice. (Pretend 5 measurements is large enough for the CLT to apply.)<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="exercises-Confidence-Intervals-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.6<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 4.3.6: "><a href="#exercises-Confidence-Intervals-3-1-2" title="Copy heading and permalink for Table 4.3.6: " aria-label="Copy heading and permalink for Table 4.3.6: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Answer 4.3.2.b.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-3-2" title="Copy heading and permalink for Answer 4.3.2.b.1" aria-label="Copy heading and permalink for Answer 4.3.2.b.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Task 4.3.2.b"><a tabindex="-1" href="#exercises-Confidence-Intervals-2-3" title="Copy heading and permalink for Task 4.3.2.b" aria-label="Copy heading and permalink for Task 4.3.2.b">🔗</a></div>
|
||||
</article><div class="autopermalink" aria-hidden="true" data-description="Exercise 4.3.2"><a tabindex="-1" href="#exercises-Confidence-Intervals-2" title="Copy heading and permalink for Exercise 4.3.2" aria-label="Copy heading and permalink for Exercise 4.3.2">🔗</a></div>
|
||||
</article>
|
||||
<article class="exercise exercise-like" id="exercises-Confidence-Intervals-3"><h3 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exercises-Confidence-Intervals-3-1-1">The weights of five mice are measured and recorded below. Give a 95% confidence interval for the sample mean weight of mice. (Pretend 5 measurements is large enough for the CLT to apply.)<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="exercises-Confidence-Intervals-3-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.6<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" aria-hidden="true" data-description="Table 4.3.6: "><a tabindex="-1" href="#exercises-Confidence-Intervals-3-1-2" title="Copy heading and permalink for Table 4.3.6: " aria-label="Copy heading and permalink for Table 4.3.6: ">🔗</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">mouse <span class="process-math">\(i\)</span>
|
||||
</th>
|
||||
@@ -532,14 +541,21 @@ p_{h} \amp = 0.4 + 1.96(0.049) \approx 0.496
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">29</td>
|
||||
</tr>
|
||||
</table></div></figure><div class="solutions"><details id="exercises-Confidence-Intervals-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-Confidence-Intervals-3-2-1"> <span class="process-math">\([23.39, 32.61]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</table></div>
|
||||
</figure><div class="solutions">
|
||||
<details id="exercises-Confidence-Intervals-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-Confidence-Intervals-3-2-1"> <span class="process-math">\([23.39, 32.61]\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-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 4.3.3.1"><a href="#exercises-Confidence-Intervals-3-2" title="Copy heading and permalink for Answer 4.3.3.1" aria-label="Copy heading and permalink for Answer 4.3.3.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 4.3.3"><a href="#exercises-Confidence-Intervals-3" title="Copy heading and permalink for Exercise 4.3.3" aria-label="Copy heading and permalink for Exercise 4.3.3">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Confidence-Intervals-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||||
<div class="para" id="exercises-Confidence-Intervals-4-1-1">The heights of five plants are measured and recorded below. Give a 95% confidence interval around the sample mean for the heights of the plants. (Pretend 5 measurements is large enough for the CLT to apply.)<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div> <figure class="table table-like" id="exercises-Confidence-Intervals-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.7<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 4.3.7: "><a href="#exercises-Confidence-Intervals-4-1-2" title="Copy heading and permalink for Table 4.3.7: " aria-label="Copy heading and permalink for Table 4.3.7: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Answer 4.3.3.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-3-2" title="Copy heading and permalink for Answer 4.3.3.1" aria-label="Copy heading and permalink for Answer 4.3.3.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Exercise 4.3.3"><a tabindex="-1" href="#exercises-Confidence-Intervals-3" title="Copy heading and permalink for Exercise 4.3.3" aria-label="Copy heading and permalink for Exercise 4.3.3">🔗</a></div>
|
||||
</article>
|
||||
<article class="exercise exercise-like" id="exercises-Confidence-Intervals-4"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
|
||||
<div class="para" id="exercises-Confidence-Intervals-4-1-1">The heights of five plants are measured and recorded below. Give a 95% confidence interval around the sample mean for the heights of the plants. (Pretend 5 measurements is large enough for the CLT to apply.)<div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</div>
|
||||
<figure class="table table-like" id="exercises-Confidence-Intervals-4-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.3.7<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" aria-hidden="true" data-description="Table 4.3.7: "><a tabindex="-1" href="#exercises-Confidence-Intervals-4-1-2" title="Copy heading and permalink for Table 4.3.7: " aria-label="Copy heading and permalink for Table 4.3.7: ">🔗</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">plant <span class="process-math">\(i\)</span>
|
||||
</th>
|
||||
@@ -565,12 +581,20 @@ p_{h} \amp = 0.4 + 1.96(0.049) \approx 0.496
|
||||
<td class="c m b0 r0 l0 t0 lines">5</td>
|
||||
<td class="c m b0 r0 l0 t0 lines">17</td>
|
||||
</tr>
|
||||
</table></div></figure><div class="solutions"><details id="exercises-Confidence-Intervals-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-Confidence-Intervals-4-2-1"> <span class="process-math">\([14.61, 19.39]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Confidence-Intervals-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||||
</table></div>
|
||||
</figure><div class="solutions">
|
||||
<details id="exercises-Confidence-Intervals-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-Confidence-Intervals-4-2-1"> <span class="process-math">\([14.61, 19.39]\text{.}\)</span><div class="autopermalink" aria-hidden="true" data-description="Paragraph"><a tabindex="-1" href="#exercises-Confidence-Intervals-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 4.3.4.1"><a tabindex="-1" href="#exercises-Confidence-Intervals-4-2" title="Copy heading and permalink for Answer 4.3.4.1" aria-label="Copy heading and permalink for Answer 4.3.4.1">🔗</a></div>
|
||||
</div></details>
|
||||
</div>
|
||||
<div class="autopermalink" aria-hidden="true" data-description="Exercise 4.3.4"><a tabindex="-1" href="#exercises-Confidence-Intervals-4" title="Copy heading and permalink for Exercise 4.3.4" aria-label="Copy heading and permalink for Exercise 4.3.4">🔗</a></div>
|
||||
</article><div class="autopermalink" aria-hidden="true" data-description="Exercises 4.3"><a tabindex="-1" href="#exercises-Confidence-Intervals" title="Copy heading and permalink for Exercises 4.3" aria-label="Copy heading and permalink for Exercises 4.3">🔗</a></div>
|
||||
</section><div class="autopermalink" aria-hidden="true" data-description="Section 4.3: Confidence Intervals"><a tabindex="-1" href="#sec-Confidence-Intervals" title="Copy heading and permalink for Section 4.3: Confidence Intervals" aria-label="Copy heading and permalink for Section 4.3: Confidence Intervals">🔗</a></div>
|
||||
</section>
|
||||
</div>
|
||||
<div class="autopermalink" data-description="Answer 4.3.4.1"><a href="#exercises-Confidence-Intervals-4-2" title="Copy heading and permalink for Answer 4.3.4.1" aria-label="Copy heading and permalink for Answer 4.3.4.1">🔗</a></div>
|
||||
</div></details></div>
|
||||
<div class="autopermalink" data-description="Exercise 4.3.4"><a href="#exercises-Confidence-Intervals-4" title="Copy heading and permalink for Exercise 4.3.4" aria-label="Copy heading and permalink for Exercise 4.3.4">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.3"><a href="#exercises-Confidence-Intervals" title="Copy heading and permalink for Exercises 4.3" aria-label="Copy heading and permalink for Exercises 4.3">🔗</a></div></section><div class="autopermalink" data-description="Section 4.3: Confidence Intervals"><a href="#sec-Confidence-Intervals" title="Copy heading and permalink for Section 4.3: Confidence Intervals" aria-label="Copy heading and permalink for Section 4.3: Confidence Intervals">🔗</a></div></section></div>
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