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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="notes-02-24"><h2 class="heading hide-type">
<span class="type">Section</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">Tuesday, Feb 24</span>
</h2>
<section class="introduction" id="notes-02-24-2"><div class="para" id="notes-02-24-2-1">This is an outline of the topics we covered in class. These notes are <em class="emphasis">not</em> a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="subsection" id="notes-02-24-3"><h3 class="heading hide-type">
<span class="type">Subsection</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">More Central Limit Theorem</span>
</h3>
<article class="example example-like" id="notes-02-24-3-2"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">119</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-24-3-2-1-1">Suppose we roll a fair D6 100 times, and let <span class="process-math">\(m\)</span> be the average of the rolls. Estimate <span class="process-math">\(\Pr(3.45 \leq m \leq 3.55)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-3-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-24-3-2-1-2">
<div class="para">Let <span class="process-math">\(R_1, \dotsc, R_{100}\)</span> be each roll’s result. Then we have previously calculated <span class="process-math">\(\E(R_i) = 3.5, \Var(R_i) = \frac{35}{12}\text{.}\)</span> We have:</div>
<div class="displaymath process-math" id="notes-02-24-3-2-1-2-3">
\begin{align*}
m \amp = \frac{R_1 + \dotsb + R_{100}}{100}. \\
\E(m) \amp = \E\left(\frac{R_1 + \dotsb + R_{100}}{100}\right) \\
\amp = \frac{1}{100}\left[\E(R_1) + \dotsb + \E(R_{100})\right] \\
\amp = \frac{1}{100}(100)(3.5) \\
\amp = 3.5 \\
\Var(m) \amp = \Var\left(\frac{R_1 + \dotsb + R_{100}}{100}\right) \\
\amp = \frac{1}{100^2} \left[\Var(R_1) + \dotsb + \Var(R_{100}) \right] \\
\amp = \frac{1}{100^2} (100)\left(\frac{35}{12}\right) \\
\amp = \frac{35}{1200}
\end{align*}
</div>
<div class="para">So, by the CLT, <span class="process-math">\(m \approx \Norm\left(3.5, \frac{35}{1200}\right)\text{.}\)</span> Even though <span class="process-math">\(m\)</span> takes on decimal values, it’s still a discrete random variable. The sum of the rolls can only take on integer values, so <span class="process-math">\(m\)</span> will take on the values <span class="process-math">\(1.00, 1.01, \dotsc, 5.99, 6.00\text{.}\)</span> Last time, we described the continuity correction as "extending the range by half a unit’s width in each direction". Here, the width of one unit is <span class="process-math">\(0.01\text{.}\)</span> So:</div>
<div class="displaymath process-math" id="notes-02-24-3-2-1-2-9">
\begin{align*}
\Pr(3.45 \leq m \leq 3.55) \amp \approx \Pr\left( \frac{3.4445 - 3.5}{\sqrt{35/1200}} \leq Z \leq \frac{3.555 - 3.5}{\sqrt{35/1200}}\right) \\
\amp \approx \Pr(-0.32 \leq Z \leq 0.32) \\
\amp = \Phi(0.32) - \Phi(-0.32) \\
\amp \approx 0.6255 - 0.3745 \\
\amp = 0.2510.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-3-2-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 119"><a href="#notes-02-24-3-2" title="Copy heading and permalink for Example 119" aria-label="Copy heading and permalink for Example 119">🔗</a></div></article><div class="autopermalink" data-description="Subsection: More Central Limit Theorem"><a href="#notes-02-24-3" title="Copy heading and permalink for Subsection: More Central Limit Theorem" aria-label="Copy heading and permalink for Subsection: More Central Limit Theorem">🔗</a></div></section><section class="subsection" id="notes-02-24-4"><h3 class="heading hide-type">
<span class="type">Subsection</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">Confidence Intervals</span>
</h3>
<div class="para logical" id="notes-02-24-4-2">
<div class="para">Idea: suppose we collected (i.i.d.) measurements <span class="process-math">\(X_1, \dotsc, X_n\)</span> which have some population mean <span class="process-math">\(\mu\)</span> and variance <span class="process-math">\(\sigma^2\text{.}\)</span> We calculate the average:</div>
<div class="displaymath process-math" id="notes-02-24-4-2-4">
\begin{gather*}
A_n = \frac{A_1 + \dotsb + A_n}{n}
\end{gather*}
</div>
<div class="para">and we want to estimate the value of <span class="process-math">\(\mu\)</span> from the collected data. Previously, we’ve discussed the MLE, the single most likely estimate of the parameter. Now, we’d like to give a range <span class="process-math">\([\mu_{\ell}, \mu_h]\)</span> where we can say something like: "we are 95% confident that <span class="process-math">\(\mu \in [\mu_{\ell}, \mu_h]\text{.}\)</span>"</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="notes-02-24-4-3">From the CLT, if <span class="process-math">\(n\)</span> is large enough, we approximate <span class="process-math">\(A_n \approx \Norm\left(\mu, \frac{\sigma^2}{n}\right)\text{.}\)</span> We can think of an interval centered at <span class="process-math">\(\mu\)</span> as a collection of numbers within a certain distance <span class="process-math">\(d\)</span> of <span class="process-math">\(\mu\text{.}\)</span> We’d like to choose a distance so that our measured <span class="process-math">\(A_n\)</span> is within <span class="process-math">\(d\)</span> of <span class="process-math">\(\mu\text{.}\)</span> We can shift our point of view here: if <span class="process-math">\(A_n\)</span> is within <span class="process-math">\(d\)</span> of <span class="process-math">\(\mu\text{,}\)</span> then <span class="process-math">\(\mu\)</span> is within <span class="process-math">\(d\)</span> of <span class="process-math">\(A_n\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="notes-02-24-4-4">TODO: image<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="notes-02-24-4-5">
<div class="para">To pick the distance <span class="process-math">\(d\text{,}\)</span> we want to make our interval large enough so that there’s only a small amount of area under the normal curve outside of the interval. So:</div>
<ol class="decimal" id="notes-02-24-4-5-2">
<li id="notes-02-24-4-5-2-1">
<div class="para" id="notes-02-24-4-5-2-1-1">Pick some value <span class="process-math">\(\alpha \in (0, 1)\)</span> (representing the area under the curve outside the interval)<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-5-2-1-1" 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="#notes-02-24-4-5-2-1" title="Copy heading and permalink for Item 1" aria-label="Copy heading and permalink for Item 1">🔗</a></div>
</li>
<li id="notes-02-24-4-5-2-2">
<div class="para" id="notes-02-24-4-5-2-2-1">Let <span class="process-math">\(c_{\alpha} = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right)\)</span> (this is the standardized <span class="process-math">\(z\)</span>-score of the right-hand side of the interval)<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-5-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 2"><a href="#notes-02-24-4-5-2-2" title="Copy heading and permalink for Item 2" aria-label="Copy heading and permalink for Item 2">🔗</a></div>
</li>
<li id="notes-02-24-4-5-2-3">
<div class="para logical" id="notes-02-24-4-5-2-3-1">
<div class="para">Then, the interval will be:</div>
<div class="displaymath process-math" id="notes-02-24-4-5-2-3-1-1">
\begin{gather*}
\left[A_n - c_{\alpha} \sqrt{\frac{\sigma^2}{n}}, A_n + c_{\alpha} \sqrt{\frac{\sigma^2}{n}}\right].
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-5-2-3-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Item 3"><a href="#notes-02-24-4-5-2-3" title="Copy heading and permalink for Item 3" aria-label="Copy heading and permalink for Item 3">🔗</a></div>
</li>
</ol>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="notes-02-24-4-6">For <span class="process-math">\(\alpha = 0.05\text{,}\)</span> we’ll have <span class="process-math">\(c_{\alpha} = \Phi^{-1}\left(1 - \frac{0.05}{2}\right) = \Phi^{-1}(0.975) = 1.96\text{.}\)</span> So:<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-6" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="theorem theorem-like" id="notes-02-24-4-7"><h4 class="heading">
<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">120</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-24-4-7-1-1">
<div class="para">The 95% confidence limits are given by:</div>
<div class="displaymath process-math" id="notes-02-24-4-7-1-1-1">
\begin{align*}
\mu_{\ell} \amp = A_n - 1.96 \sqrt{\frac{\sigma^2}{n}} \\
\mu_{h} \amp = A_n + 1.96 \sqrt{\frac{\sigma^2}{n}}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Theorem 120"><a href="#notes-02-24-4-7" title="Copy heading and permalink for Theorem 120" aria-label="Copy heading and permalink for Theorem 120">🔗</a></div></article><article class="definition definition-like" id="notes-02-24-4-8"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">121</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-24-4-8-1-1">The term <span class="process-math">\(\sqrt{\frac{\sigma^2}{n}}\)</span> is called the <dfn class="terminology">standard error of the mean</dfn>. (It’s the standard deviation of <span class="process-math">\(A_n\text{.}\)</span>)<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 121"><a href="#notes-02-24-4-8" title="Copy heading and permalink for Definition 121" aria-label="Copy heading and permalink for Definition 121">🔗</a></div></article><div class="para" id="notes-02-24-4-9">We don’t know the true value of <span class="process-math">\(\sigma^2\text{,}\)</span> just like we don’t know the true value of <span class="process-math">\(\mu\text{.}\)</span> So we’ll have to use our sample of measurements to estimate <span class="process-math">\(\sigma^2\text{,}\)</span> and then use that estimate to give us our range of values for <span class="process-math">\(\mu\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-9" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="definition definition-like" id="notes-02-24-4-10"><h4 class="heading">
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">122</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para logical" id="notes-02-24-4-10-1-1">
<div class="para">Let <span class="process-math">\(X_1, \dotsc, X_n\)</span> be (i.i.d.) measurements. Then the <dfn class="terminology">sample mean</dfn> is:</div>
<div class="displaymath process-math" id="notes-02-24-4-10-1-1-3">
\begin{gather*}
A_n = \frac{X_1 + \dotsb + X_n}{n}
\end{gather*}
</div>
<div class="para">and the <dfn class="terminology">sample variance</dfn> is:</div>
<div class="displaymath process-math" id="notes-02-24-4-10-1-1-5">
\begin{gather*}
s^2 = \frac{\sum (X_i - A_n)^2}{n - 1} = \frac{\left(\sum X_i^2\right) - n(A_n^2)}{n - 1}
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-10-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Definition 122"><a href="#notes-02-24-4-10" title="Copy heading and permalink for Definition 122" aria-label="Copy heading and permalink for Definition 122">🔗</a></div></article><div class="para" id="notes-02-24-4-11">Why <span class="process-math">\(n - 1\)</span> here? It turns out that, as defined above, <span class="process-math">\(s^2\)</span> is an unbiased estimator for <span class="process-math">\(\sigma^2\text{,}\)</span> whereas it wouldn’t be if we divided by <span class="process-math">\(n\)</span> instead. (We’ll ommit both the calculation justifying this and a more conceptual explanation for now.)<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-11" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="notes-02-24-4-12">
<div class="para">So, we can adjust the confidence limits from the theorem:</div>
<div class="displaymath process-math" id="notes-02-24-4-12-1">
\begin{align*}
\mu_{\ell} \amp = A_n - 1.96 \sqrt{\frac{s^2}{n}} \\
\mu_{h} \amp = A_n + 1.96 \sqrt{\frac{s^2}{n}}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-12" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="notes-02-24-4-13"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">123</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-24-4-13-1-1">Suppose we’re given measurements <span class="process-math">\(X_1, X_2, X_3, X_4, X_5\)</span> below.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(i\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X_i\)</span></th>
</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">\(19.2\)</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">\(20.1\)</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">\(21.3\)</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">\(20.7\)</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">\(19.8\)</span></td>
</tr>
</table></div> <div class="para logical" id="notes-02-24-4-13-1-3">
<div class="para">We’re just practicing the computation here, so we’ll pretend that 5 measurements is large enough for the CLT to apply.</div>
<div class="displaymath process-math" id="notes-02-24-4-13-1-3-1">
\begin{align*}
A_n \amp = \frac{19.2 + \dotsb + 19.8}{5} = 20.22 \\
\sum X_i^2 \amp = 19.2^2 + \dotsb + 19.8^2 = 2046.87 \\
s^2 \amp = \frac{2046.87 - 5(20.22^2)}{4} = 0.657 \\
\sqrt{\frac{s^2}{n}} \amp = \sqrt{\frac{0.657}{5}} \approx 0.362 \\
\mu_{\ell} \amp = 20.22 - 1.96(0.362) \approx 19.51 \\
\mu_{h} \amp = 20.22 + 1.96(0.362) \approx 20.93
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-13-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Example 123"><a href="#notes-02-24-4-13" title="Copy heading and permalink for Example 123" aria-label="Copy heading and permalink for Example 123">🔗</a></div></article><div class="para" id="notes-02-24-4-14">This way of computing confidence limits only applies if we can think of the parameter we’re estimating as a mean.<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-14" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<article class="example example-like" id="notes-02-24-4-15"><h4 class="heading">
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">124</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
</h4>
<div class="para" id="notes-02-24-4-15-1-1">Suppose we flip a coin 100 times and see 40 heads. Find a 98% confidence interval for the bias <span class="process-math">\(p\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-15-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-24-4-15-1-2">
<div class="para">Let <span class="process-math">\(H_1, \dotsc, H_{100}\)</span> indicate heads on each flip. Let <span class="process-math">\(S_{100} = H_1 + \dotsb + H_{100}\)</span> and <span class="process-math">\(A_n = \frac{S_n}{n}\text{.}\)</span> Then:</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-2-4">
\begin{align*}
\E(A_n) \amp = \E\left(\frac{S_n}{n}\right) = \frac{1}{n}\E(S_n) = \frac{1}{n} np = p
\end{align*}
</div>
<div class="para">So a 98% confidence interval for <span class="process-math">\(p\)</span> is also a 98% confidence interval for <span class="process-math">\(p\text{.}\)</span>
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-15-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-24-4-15-1-3">
<div class="para">In this setting, we can simplify the computation of <span class="process-math">\(s^2\)</span> and <span class="process-math">\(\sqrt{\frac{s^2}{n}}\text{,}\)</span> using our knowledge of the binomial distribution <span class="process-math">\(S_n\text{.}\)</span>
</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-3-4">
\begin{align*}
\Var(S_n) \amp = np(1 - p) \\
\Var(A_n) \amp = \Var\left(\frac{S_n}{n}\right) = \frac{1}{n^2} \Var(S_n) = \frac{1}{n^2} np(1-p) = \frac{p(1-p)}{n}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(s^2 = p(1 - p)\)</span> and <span class="process-math">\(\sqrt{\frac{s^2}{n}} = \sqrt{\frac{p(1-p)}{n}}\text{.}\)</span> We don’t know the value of <span class="process-math">\(p\text{,}\)</span> so we’ll use the MLE <span class="process-math">\(\widehat{p} = \frac{40}{100} = 0.4\text{.}\)</span> So:</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-3-9">
\begin{gather*}
\sqrt{\frac{s^2}{n}} = \sqrt{\frac{(0.4)(0.6)}{100}} \approx 0.049.
\end{gather*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#notes-02-24-4-15-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para logical" id="notes-02-24-4-15-1-4">
<div class="para">For a 98% confidence interval, we have <span class="process-math">\(\alpha = 1 - 0.98 = 0.02\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-4-2">
\begin{gather*}
c_{\alpha} = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) = \Phi^{-1}(0.99) \approx 2.33.
\end{gather*}
</div>
<div class="para">Then, the confidence limits are:</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-4-3">
\begin{align*}
p_{\ell} \amp = 0.4 - 2.33(0.049) \approx 0.286 \\
p_{h} \amp = 0.4 + 2.33(0.049) \approx 0.514
\end{align*}
</div>
<div class="para">For comparison, the 95% confidence limits are:</div>
<div class="displaymath process-math" id="notes-02-24-4-15-1-4-4">
\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="#notes-02-24-4-15-1-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
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