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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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<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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<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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<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>
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<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>
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<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>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<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>
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<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>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<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>
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<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>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</span></a></div>
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<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>
|
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</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>
|
||
</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">
|
||
<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>
|
||
</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>
|
||
</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>
|
||
</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>
|
||
</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">
|
||
<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>
|
||
</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>
|
||
</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">
|
||
<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>
|
||
</li>
|
||
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="title">Colophon</span></a></div></li>
|
||
</ul>
|
||
</li>
|
||
</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">
|
||
<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>
|
||
</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>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/thm-CLT.html" id="sec-Confidence-Intervals-3-4">
|
||
\begin{gather*}
|
||
A_n = \frac{X_1 + \dotsb + X_n}{n}.
|
||
\end{gather*}
|
||
</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>
|
||
<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>
|
||
<article class="definition definition-like" id="def-sample-variance"><h3 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>
|
||
<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">
|
||
\begin{gather*}
|
||
s^2 = \frac{\sum \left(X_i - A_n\right)^2}{n - 1} = \frac{\left(\sum X_i^2\right) - n \left(A_n\right)^2}{n - 1}.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">The expression:</div>
|
||
<div class="displaymath process-math" id="def-sample-variance-1-1-5">
|
||
\begin{gather*}
|
||
\sqrt{\frac{s^2}{n}} = \frac{s}{\sqrt{n}}
|
||
\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>
|
||
<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>
|
||
<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>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-7-7">
|
||
\begin{gather*}
|
||
\left[A_n - c_{\alpha} \left(\frac{s}{\sqrt{n}}\right), A_n + c_{\alpha} \left(\frac{s}{\sqrt{n}}\right)\right]
|
||
\end{gather*}
|
||
</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>
|
||
<article class="theorem theorem-like" id="thm-CI"><h3 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>
|
||
<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">
|
||
\begin{align*}
|
||
\mu_{\ell} \amp = A_n - 1.96 \left(\frac{s}{\sqrt{n}}\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>
|
||
<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">
|
||
<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">
|
||
<tr>
|
||
<td class="c m b1 r0 l0 t0 lines">Plant <span class="process-math">\(i\)</span>
|
||
</td>
|
||
<td class="c m b1 r0 l0 t0 lines">Height <span class="process-math">\(H_i\)</span> (in)</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(9.2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(9.8\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(10.1\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4\)</span></td>
|
||
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(9.5\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<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">
|
||
<div class="para">Then the sample mean is:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-1">
|
||
\begin{align*}
|
||
A_5 = \frac{H_1 + \dotsb + H_5}{5} \amp = \frac{9.2 + \dotsb + 10.4}{5} = 9.8.
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">The sum of the squares is:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-2">
|
||
\begin{align*}
|
||
\sum H_i^2 \amp = H_1^2 + \dotsb + H_5^2 = 9.2^2 + \dotsb + 10.4^2 = 481.1
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">So the sample variance is:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-3">
|
||
\begin{gather*}
|
||
s^2 = \frac{\left(\sum H_i\right)^2 - 5 (A_5)^2}{5 - 1} = \frac{481.1 - 5(9.8)^2}{4} \approx 0.225
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Now, the standard error of the mean is:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-4">
|
||
\begin{gather*}
|
||
\frac{s}{\sqrt{n}} = \sqrt{\frac{0.225}{5}} \approx 0.212
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Finally, we can give 95% confidence limits:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-9-1-3-5">
|
||
\begin{align*}
|
||
\mu_{\ell} \amp = 9.8 - 1.96(0.212) \approx 9.38 \\
|
||
\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>
|
||
<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="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*}
|
||
\Var(S_n) \amp = np(1 - p) \\
|
||
\Var(A_n) \amp = \Var\left(\frac{S_n}{n}\right) \\
|
||
\amp = \frac{1}{n^2} \Var\left(S_n\right) \\
|
||
\amp = \frac{1}{n^2} (np(1 - p)) \\
|
||
\amp = \frac{p(1 - p)}{n}.
|
||
\end{align*}
|
||
</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>
|
||
<article class="example example-like" id="sec-Confidence-Intervals-11"><h3 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>
|
||
<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">
|
||
\begin{gather*}
|
||
\sqrt{\frac{(0.4)(1 - 0.4)}{100}} = \sqrt{\frac{0.24}{100}} \approx 0.049.
|
||
\end{gather*}
|
||
</div>
|
||
<div class="para">Therefore, 95% confidence limits for <span class="process-math">\(p\)</span> are:</div>
|
||
<div class="displaymath process-math" id="sec-Confidence-Intervals-11-1-1-5">
|
||
\begin{align*}
|
||
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>
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||
</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">
|
||
<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>
|
||
</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="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>
|
||
</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>
|
||
<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>
|
||
</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>
|
||
<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>
|
||
</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">
|
||
<tr class="header-horizontal">
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines">mouse <span class="process-math">\(i\)</span>
|
||
</th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines">weight <span class="process-math">\(W_i\)</span> (g)</th>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||
<td class="c m b0 r0 l0 t0 lines">26</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines">32</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">33</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">20</td>
|
||
</tr>
|
||
<tr>
|
||
<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>
|
||
</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">
|
||
<tr class="header-horizontal">
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines">plant <span class="process-math">\(i\)</span>
|
||
</th>
|
||
<th scope="col" class="c m b1 r0 l0 t0 lines">height <span class="process-math">\(H_i\)</span> (in)</th>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">1</td>
|
||
<td class="c m b0 r0 l0 t0 lines">15</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">2</td>
|
||
<td class="c m b0 r0 l0 t0 lines">14</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">18</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r0 l0 t0 lines">4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">21</td>
|
||
</tr>
|
||
<tr>
|
||
<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>
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</div>
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<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>
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</div></details></div>
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<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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