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2026-04-09 19:54:49 -04:00
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@@ -414,7 +414,7 @@ eBookConfig.enable_chatcodes = false;
<div class="solutions"><details id="exam-02-5-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="exam-02-5-1-2-1">
<div class="para">
<span class="process-math">\(S\)</span> is binomially distributed with <span class="process-math">\(E(S) = (180)(0.7) = 126\)</span> and <span class="process-math">\(\Var(S) = (180)(0.7)(0.3) = 37.8\text{.}\)</span> Therefore:</div>
<span class="process-math">\(S\)</span> is binomially distributed with <span class="process-math">\(\E(S) = (180)(0.7) = 126\)</span> and <span class="process-math">\(\Var(S) = (180)(0.7)(0.3) = 37.8\text{.}\)</span> Therefore:</div>
<div class="displaymath process-math" id="exam-02-5-1-2-1-4">
\begin{align*}
\Pr(120 \leq S \leq 130) \amp \approx \Pr\left( \frac{119.5 - 126}{\sqrt{37.8}} \leq Z \leq \frac{130.5 - 126}{\sqrt{37.8}}\right) \\
@@ -431,7 +431,7 @@ eBookConfig.enable_chatcodes = false;
</div></details></div>
<div class="workspace" data-space="1in"></div>
<div class="autopermalink" data-description="Worksheet Exercise 3"><a href="#exam-02-5-1" title="Copy heading and permalink for Worksheet Exercise 3" aria-label="Copy heading and permalink for Worksheet Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="exam-02-5-2"><h3 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="exam-02-5-2-1-1">The weights of five apples are measured and recorded below. Find the sample mean, sample variance, and a 95\% confidence interval around the sample mean for the weights of the apples. (Pretend that 5 measurements is enough for the CLT to apply. Do not use the <span class="process-math">\(t\)</span>-distribution.)<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
<div class="para" id="exam-02-5-2-1-1">The weights of five apples are measured and recorded below. Find the sample mean, sample variance, and a 95% confidence interval around the sample mean for the weights of the apples. (Pretend that 5 measurements is enough for the CLT to apply. Do not use the <span class="process-math">\(t\)</span>-distribution.)<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-5-2-1-1" 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>
<td class="c m b1 r1 l0 t0 lines">Apple <span class="process-math">\(i\)</span>
@@ -528,11 +528,11 @@ k = 64.5 + 1.65 \sqrt{38.4} \amp \approx 74.72
<div class="para">If the coin is fair, then <span class="process-math">\(\E(S) = (160)(0.5) = 80\)</span> and <span class="process-math">\(\Var(S) = (160)(0.5)(0.5) = 40\text{.}\)</span> The probability of rejecting <span class="process-math">\(H_0\)</span> is then:</div>
<div class="displaymath process-math" id="exam-02-7-1-3-2-1-4">
\begin{align*}
\Pr(S \geq 75) \amp \approx \Pr\left(Z \geq \frac{75 - 80}{\sqrt{40}}\right) \\
\amp \approx \Pr(Z \geq -0.79) \\
\amp \approx 1 - \Phi(-0.79) \\
\amp \approx 1 - 0.2148 \\
\amp = 0.7852.
\Pr(S \geq 75) \amp \approx \Pr\left(Z \geq \frac{74.5 - 80}{\sqrt{40}}\right) \\
\amp \approx \Pr(Z \geq -0.87) \\
\amp \approx 1 - \Phi(-0.87) \\
\amp \approx 1 - 0.1922 \\
\amp = 0.8078.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#exam-02-7-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>