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@@ -101,14 +101,14 @@ eBookConfig.allow_pairs = false;
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eBookConfig.python3 = null;
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@@ -146,6 +146,7 @@ eBookConfig.enable_chatcodes = false;
\DeclareMathOperator{\Geom}{Geom}
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\DeclareMathOperator{\Exp}{Exp}
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@@ -294,12 +295,156 @@ eBookConfig.enable_chatcodes = false;
</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></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-CLT"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Central Limit Theorem</span>
</h2>
<div class="para" id="sec-CLT-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="sec-CLT-2">Although we defined the normal distribution earlier (<a href="sec-Continuous-RVs.html#def-normal-distribution" class="xref" data-knowl="./knowl/xref/def-normal-distribution.html" data-reveal-label="Reveal" data-close-label="Close" title="Definition 2.2.8">Definition 2.2.8</a>), we haven’t done much in the way of probability calculations with it. The sense in which this distribution is "normal" is that it turns out other distributions, when added together or averaged, tend to be well-approximated by a normal distribution.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="theorem theorem-like" id="thm-CLT"><h3 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="thm-CLT-1-1">
<div class="para">Suppose <span class="process-math">\(X_1, X_2, \dotsc, X_n\)</span> are <dfn class="terminology">independent and identically distributed</dfn> (or <dfn class="terminology">iid</dfn>) random variables with finite expected value <span class="process-math">\(\mu\)</span> and finite variance <span class="process-math">\(\sigma^2\text{.}\)</span> Let:</div>
<div class="displaymath process-math" id="thm-CLT-1-1-6">
\begin{align*}
S_n \amp = X_1 + X_2 + \dotsb + X_n \\
A_n \amp = \frac{X_1 + X_2 + \dotsb + X_n}{n}
\end{align*}
</div>
<div class="para">Then, for sufficiently large <span class="process-math">\(n\text{:}\)</span>
</div>
<div class="displaymath process-math" id="thm-CLT-1-1-8">
\begin{gather*}
S_n \approx \Norm(n \mu, n \sigma^2) \\
A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#thm-CLT-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Theorem 4.2.1"><a href="#thm-CLT" title="Copy heading and permalink for Theorem 4.2.1" aria-label="Copy heading and permalink for Theorem 4.2.1">🔗</a></div></article><article class="remark remark-like" id="sec-CLT-4"><h3 class="heading">
<span class="type">Remark</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="sec-CLT-4-1">
<div class="para">How large is "sufficiently large"? We’ll make use of two common rules of thumb:</div>
<ol class="decimal" id="sec-CLT-4-1-1"><li id="sec-CLT-4-1-1-1">
<div class="para" id="sec-CLT-4-1-1-1-1">For general distributions, we’d like <span class="process-math">\(n \geq 30\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 1"><a href="#sec-CLT-4-1-1-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
</li></ol>
<ol class="decimal" id="sec-CLT-4-1-2"><li id="sec-CLT-4-1-2-1">
<div class="para" id="sec-CLT-4-1-2-1-1">For a binomial distribution specifically, we’d like <span class="process-math">\(n\)</span> large enough that we have at least five heads and five tails.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 1"><a href="#sec-CLT-4-1-2-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
</li></ol>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-4-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Remark 4.2.2"><a href="#sec-CLT-4" title="Copy heading and permalink for Remark 4.2.2" aria-label="Copy heading and permalink for Remark 4.2.2">🔗</a></div></article><div class="para logical" id="sec-CLT-5">
<div class="para">When a normal distribution gets translated and/or scaled, the result is another normal distribution. It will often be useful to <em class="emphasis">standardize</em> a normal distribution <span class="process-math">\(X = \Norm(\mu, \sigma^2)\)</span> as follows:</div>
<div class="displaymath process-math" id="sec-CLT-5-3">
\begin{gather*}
Z = \frac{X - \mu}{\sigma} = \Norm(0, 1).
\end{gather*}
</div>
<div class="para">Given an <span class="process-math">\(X\)</span>-value <span class="process-math">\(a\text{,}\)</span> the corresponding value of <span class="process-math">\(Z\)</span> is called a <dfn class="terminology"><span class="process-math">\(z\)</span>-score</dfn>, and is sometimes denoted <span class="process-math">\(a^*\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-CLT-6"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.3</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="sec-CLT-6-1-1">
<div class="para">Suppose we flip a fair coin 100 times and let <span class="process-math">\(S_{100}\)</span> be the number of heads. In order to calculate, for example, <span class="process-math">\(\Pr(40 \leq S_{100} \leq 60)\)</span> using the binomial distribution probability formula, we would need to calculate:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/thm-CLT.html" id="sec-CLT-6-1-1-3">
\begin{gather*}
\Pr(40 \leq S_{100} \leq 60) = \sum_{k = 40}^{60} {100 \choose k} (0.5)^k (1 - 0.5)^{100 - k}.
\end{gather*}
</div>
<div class="para">This could be done using a good calculator. If we increased the number of flips to 1000, or 10000, etc., even a powerful calculator might struggle with the computation. However, if we’re willing to accept an approximation, then <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> gives a (mostly) straightforward path:</div>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/thm-CLT.html" id="sec-CLT-6-1-1-5">
\begin{align*}
\E(S_{100}) \amp = np = (100)(0.5) = 50 \\
\Var(S_{100}) \amp = np(1-p) = (100)(0.5)(0.5) = 25 \\
\Rightarrow \quad S_{100} \amp\approx \Norm(50, 25) \\
Z \amp = \frac{S_{100} - 50}{\sqrt{25}} \\
\Pr(40 \leq S_{100} \leq 60) \amp\approx \Pr\left(\frac{40 - 50}{5} \leq Z \leq \frac{60 - 50}{5}\right) \\
\amp= \Pr\left(-2 \leq Z \leq 2\right)
\end{align*}
</div>
<div class="para">We now reach a potential impass.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 4.2.3"><a href="#sec-CLT-6" title="Copy heading and permalink for Example 4.2.3" aria-label="Copy heading and permalink for Example 4.2.3">🔗</a></div></article><div class="para logical" id="sec-CLT-7">
<div class="para">The random variable <span class="process-math">\(Z\)</span> is the <em class="emphasis">standard normal distribution</em>, with pdf:</div>
<div class="displaymath process-math" id="sec-CLT-7-3">
\begin{gather*}
\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}
\end{gather*}
</div>
<div class="para">To calculate probabilities like <span class="process-math">\(\Pr(-2 \leq Z \leq 2)\text{,}\)</span> we should evaluate the integral:</div>
<div class="displaymath process-math" id="sec-CLT-7-5">
\begin{gather*}
\Pr(-2 \leq Z \leq 2) = \int_{-2}^2 \phi(x)\ dx = \int_{-2}^2 \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\ dx
\end{gather*}
</div>
<div class="para">Alternatively, we could use the cdf:</div>
<div class="displaymath process-math" id="sec-CLT-7-6">
\begin{gather*}
\Phi(x) = \int_{-\infty}^x \phi(t)\ dt = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-t^2/2}\ dt
\end{gather*}
</div>
<div class="para">If we had a nice formula for <span class="process-math">\(\Phi(x)\text{,}\)</span> then we could complete the probability calculation:</div>
<div class="displaymath process-math" id="sec-CLT-7-8">
\begin{gather*}
\Pr(-2\leq Z \leq 2) = \Phi(2) - \Phi(-2).
\end{gather*}
</div>
<div class="para">Unfortunately, we don’t have a nice formula for <span class="process-math">\(\Phi(x)\text{.}\)</span> In fact, there is provably no "nice" formula that could be written for <span class="process-math">\(\Phi(x)\)</span> (where "nice" has some technical mathematical meaning that we won’t discuss here).</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="sec-CLT-8">The true power of <a href="sec-CLT.html#thm-CLT" class="xref" data-knowl="./knowl/xref/thm-CLT.html" data-reveal-label="Reveal" data-close-label="Close" title="Theorem 4.2.1">Theorem 4.2.1</a> comes to the rescue: it doesn’t matter what type of distribution the individual <span class="process-math">\(X_i\)</span>’s have, and it doesn’t matter what type of distributions <span class="process-math">\(S_n\)</span> and <span class="process-math">\(A_n\)</span> have. The random variable <span class="process-math">\(Z\)</span> is <span class="process-math">\(\Norm(0, 1)\text{.}\)</span> Since it’s the same particular distribution all the time, and since we’re only aiming at an approximation anyway, we can calculate many estimations of probabilities like <span class="process-math">\(\Pr(a \leq Z \leq b)\)</span> ahead of time and just save the results to look up later.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-8" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="sec-CLT-9">A table of precalculated <span class="process-math">\(\Phi(z)\)</span> values can be found in <a href="app-Phi-table.html" class="internal" title="Section A.1: \Phi(z) Table of Values">Section A.1</a>.<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-9" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="sec-CLT-10"><h3 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">4.2.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h3>
<div class="para logical" id="sec-CLT-10-1-1">
<div class="para">Continuing the previous example, and having looked up the values <span class="process-math">\(\Phi(2) \approx 0.9772\)</span> and <span class="process-math">\(\Phi(-2) \approx 0.0228\text{,}\)</span> we complete our estimation:</div>
<div class="displaymath process-math" id="sec-CLT-10-1-1-3">
\begin{align*}
\Pr(40 \leq S_{100} \leq 60) \amp\approx \Pr\left(\frac{40 - 50}{5} \leq Z \leq \frac{60 - 50}{5}\right) \\
\amp= \Pr\left(-2 \leq Z \leq 2\right) \\
\amp \approx \Phi(2) - \Phi(-2) \\
\amp \approx 0.9772 - 0.0228 \\
\amp = 0.9544.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 4.2.4"><a href="#sec-CLT-10" title="Copy heading and permalink for Example 4.2.4" aria-label="Copy heading and permalink for Example 4.2.4">🔗</a></div></article><div class="para logical" id="sec-CLT-11">
<div class="para">The true value is approximately <span class="process-math">\(0.965\text{,}\)</span> so we underestimated the probability a bit. It turns out this will consistently be the case when using a normal approximation to a discrete distribution, such as the binomial distribution. The range of values <span class="process-math">\(40 \leq S_{100} \leq 60\)</span> is a list of <span class="process-math">\(21\)</span> discrete values. However, the continuous interval <span class="process-math">\([40, 60]\)</span> has a length of only 20. So calculating the <span class="process-math">\(z\)</span>-scores from the values <span class="process-math">\(40\)</span> and <span class="process-math">\(60\)</span> has the effect of shortening the interval length by 1. To mitigate this, we can make a <dfn class="terminology">continuity correction</dfn> by extending the values out by <span class="process-math">\(0.5\)</span> in each direction:</div>
<div class="displaymath process-math" id="sec-CLT-11-10">
\begin{align*}
\Pr(40 \leq S_{100} \leq 60) \amp\approx \Pr\left(\frac{39.5 - 50}{5} \leq Z \leq \frac{60.5 - 50}{5}\right) \\
\amp= \Pr\left(-2.1 \leq Z \leq 2.1\right) \\
\amp \approx \Phi(2.1) - \Phi(-2.1) \\
\amp \approx 0.9821 - 0.0179 \\
\amp = 0.9642.
\end{align*}
</div>
<div class="para">This is a much more accurate estimate.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#sec-CLT-11" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<section class="exercises" id="exercises-CLT"><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>
@@ -307,21 +452,51 @@ eBookConfig.enable_chatcodes = false;
<article class="exercise exercise-like" id="exercises-CLT-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-CLT-1-1-1">Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. Let <span class="process-math">\(S\)</span> be the number of heads. Estimate the probability that <span class="process-math">\(34 \leq S \leq 44\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-CLT-1-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-CLT-1-2-1">
<span class="process-math">\(\Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 4.2.1.1"><a href="#exercises-CLT-1-2" title="Copy heading and permalink for Answer 4.2.1.1" aria-label="Copy heading and permalink for Answer 4.2.1.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 4.2.1"><a href="#exercises-CLT-1" title="Copy heading and permalink for Exercise 4.2.1" aria-label="Copy heading and permalink for Exercise 4.2.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-CLT-2-1-1">Suppose a fair die is rolled 100 times, and let <span class="process-math">\(m\)</span> be the average value of the rolls. Estimate the probability that <span class="process-math">\(3.45 \leq m \leq 3.55\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="exercises-CLT-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-CLT-2-2-1">
<span class="process-math">\(\Pr(3.45 \leq m \leq 3.55) \approx 0.6141 - 0.3859 = 0.2282.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 4.2.2.1"><a href="#exercises-CLT-2-2" title="Copy heading and permalink for Answer 4.2.2.1" aria-label="Copy heading and permalink for Answer 4.2.2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 4.2.2"><a href="#exercises-CLT-2" title="Copy heading and permalink for Exercise 4.2.2" aria-label="Copy heading and permalink for Exercise 4.2.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-CLT-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="exercises-CLT-3-1"><div class="para" id="exercises-CLT-3-1-1">The heights of men in the US have a mean of 69 in and a variance of about 9 in<span class="process-math">\(^2\text{,}\)</span> and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in<span class="process-math">\(^2\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="task exercise-like" id="exercises-CLT-3-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="exercises-CLT-3-2-1-1">Suppose the heights of 30 men are sampled, and a sample mean <span class="process-math">\(m\)</span> is taken. Find the expected value and variance of <span class="process-math">\(m\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exercises-CLT-3-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-CLT-3-2-2-1">
<span class="process-math">\(\E(m) = 69\)</span> and <span class="process-math">\(\Var(m) = 0.3\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 4.2.3.a.1"><a href="#exercises-CLT-3-2-2" title="Copy heading and permalink for Answer 4.2.3.a.1" aria-label="Copy heading and permalink for Answer 4.2.3.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 4.2.3.a"><a href="#exercises-CLT-3-2" title="Copy heading and permalink for Task 4.2.3.a" aria-label="Copy heading and permalink for Task 4.2.3.a">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="exercises-CLT-3-3-1-1">Estimate the probability that <span class="process-math">\(m \geq 69.5\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 4.2.3.b"><a href="#exercises-CLT-3-3" title="Copy heading and permalink for Task 4.2.3.b" aria-label="Copy heading and permalink for Task 4.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-CLT-3-4-1-1">What if the sampled group was women?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="solutions"><details id="exercises-CLT-3-3-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-CLT-3-3-2-1">
<span class="process-math">\(\Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 4.2.3.b.1"><a href="#exercises-CLT-3-3-2" title="Copy heading and permalink for Answer 4.2.3.b.1" aria-label="Copy heading and permalink for Answer 4.2.3.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 4.2.3.b"><a href="#exercises-CLT-3-3" title="Copy heading and permalink for Task 4.2.3.b" aria-label="Copy heading and permalink for Task 4.2.3.b">🔗</a></div></article><article class="task exercise-like" id="exercises-CLT-3-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="exercises-CLT-3-4-1-1">If the sampled group was women, estimate the probability that <span class="process-math">\(m \geq 64\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="exercises-CLT-3-4-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
<div class="para" id="exercises-CLT-3-4-2-1">
<span class="process-math">\(\E(m) = 63.5\)</span> and <span class="process-math">\(\Var(m) \approx 0.208\text{.}\)</span> <span class="process-math">\(\Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-CLT-3-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Answer 4.2.3.c.1"><a href="#exercises-CLT-3-4-2" title="Copy heading and permalink for Answer 4.2.3.c.1" aria-label="Copy heading and permalink for Answer 4.2.3.c.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 4.2.3.c"><a href="#exercises-CLT-3-4" title="Copy heading and permalink for Task 4.2.3.c" aria-label="Copy heading and permalink for Task 4.2.3.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 4.2.3"><a href="#exercises-CLT-3" title="Copy heading and permalink for Exercise 4.2.3" aria-label="Copy heading and permalink for Exercise 4.2.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 4.2"><a href="#exercises-CLT" title="Copy heading and permalink for Exercises 4.2" aria-label="Copy heading and permalink for Exercises 4.2">🔗</a></div></section><div class="autopermalink" data-description="Section 4.2: Central Limit Theorem"><a href="#sec-CLT" title="Copy heading and permalink for Section 4.2: Central Limit Theorem" aria-label="Copy heading and permalink for Section 4.2: Central Limit Theorem">🔗</a></div></section></div>
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