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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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<section class="introduction" id="quiz-04-2"><div class="para" id="quiz-04-2-1">The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="quiz-04-2-2">Show all work unless instructed otherwise.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="onepage" id="quiz-04-3"><article class="exercise exercise-like" id="quiz-04-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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<div class="para" id="quiz-04-3-1-1-1">Suppose we have a coin that we’re told is not fair and has a probability of <span class="process-math">\(\theta = 0.6\)</span> of coming up heads. We suspect the probability is even higher than this.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="para" id="quiz-04-3-1-1-2">Throughout the following parts, do <em class="emphasis">not</em> use a normal approximation.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-3-1-1-2" 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="quiz-04-3-1-2"><h4 class="heading"><span class="codenumber">(a)</span></h4>
<div class="para" id="quiz-04-3-1-2-1-1">We flip the coin 10 times and see 9 heads. Should we use a 1-tailed test or a 2-tailed test? Find the <span class="process-math">\(p\)</span>-value. Do we have strong enough evidence to reject the null hypothesis at a 0.05 significance level?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-3-1-2-1-1" 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="quiz-04-3-1-2-2" 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="quiz-04-3-1-2-2-1">
<div class="para">Using a 1-tailed test, the <span class="process-math">\(p\)</span>-value is <span class="process-math">\(\Pr( \geq 9 \text{ heads})\text{:}\)</span>
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<div class="displaymath process-math" id="quiz-04-3-1-2-2-1-3">
\begin{align*}
\Pr(9 \text{ heads}) \amp = {10 \choose 9} (0.6)^9 (1 - 0.6)^{10 - 9} = 10(0.6)^9(0.4)^1 \approx 0.04 \\
\Pr(10 \text{ heads}) \amp = {10 \choose 10} (0.6)^{10} (1 - 0.6)^{10 - 10} = 1(0.6)^{10}(0.4)^{0} \approx 0.006
\end{align*}
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<div class="para">The <span class="process-math">\(p\)</span>-value is the sum: <span class="process-math">\(\Pr(\geq 9 \text{ heads}) \approx 0.046 \lt 0.05\text{,}\)</span> so we can reject the null hypothesis.</div>
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<div class="autopermalink" data-description="Solution 1.a.1"><a href="#quiz-04-3-1-2-2" title="Copy heading and permalink for Solution 1.a.1" aria-label="Copy heading and permalink for Solution 1.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.a"><a href="#quiz-04-3-1-2" title="Copy heading and permalink for Task 1.a" aria-label="Copy heading and permalink for Task 1.a">🔗</a></div></article><article class="task exercise-like" id="quiz-04-3-1-3"><h4 class="heading"><span class="codenumber">(b)</span></h4>
<div class="para" id="quiz-04-3-1-3-1-1">What is the minimum number of heads we would need to see in 10 flips to reject the null hypothesis?<div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-3-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="solutions"><details id="quiz-04-3-1-3-2" 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="quiz-04-3-1-3-2-1">
<div class="para">We saw in part (a) that at least 9 heads is enough to reject. Then:</div>
<div class="displaymath process-math" id="quiz-04-3-1-3-2-1-1">
\begin{gather*}
\Pr(8 \text{ heads}) = {10 \choose 8} (0.6)^8 (1 - 0.6)^{10 - 8} = 45(0.6)^8(0.4)^2 \approx 0.121
\end{gather*}
</div>
<div class="para">Adding to the previous probabilities, <span class="process-math">\(\Pr(\geq 8 \text{ heads}) \approx 0.046 + 0.121 = 0.167 \gt 0.05\text{.}\)</span> So 8 heads would not be enough to reject the null hypothesis, therefore the minimum number of heads to reject would be 9.</div>
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<div class="autopermalink" data-description="Solution 1.b.1"><a href="#quiz-04-3-1-3-2" title="Copy heading and permalink for Solution 1.b.1" aria-label="Copy heading and permalink for Solution 1.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.b"><a href="#quiz-04-3-1-3" title="Copy heading and permalink for Task 1.b" aria-label="Copy heading and permalink for Task 1.b">🔗</a></div></article><article class="task exercise-like" id="quiz-04-3-1-4"><h4 class="heading"><span class="codenumber">(c)</span></h4>
<div class="para" id="quiz-04-3-1-4-1-1">What is the power of the test if the true value of the parameter is <span class="process-math">\(\theta = 0.8\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-04-3-1-4-1-1" 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="quiz-04-3-1-4-2" 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="quiz-04-3-1-4-2-1">
<div class="para">With <span class="process-math">\(\theta = 0.8\text{:}\)</span>
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<div class="displaymath process-math" id="quiz-04-3-1-4-2-1-2">
\begin{align*}
\Pr(9 \text{ heads}) \amp = {10 \choose 9} (0.8)^9 (1 - 0.8)^{10 - 9} = 10(0.8)^9(0.2)^1 \approx 0.268 \\
\Pr(10 \text{ heads}) \amp = {10 \choose 10} (0.8)^{10} (1 - 0.8)^{10 - 10} = 1(0.8)^{10}(0.2)^{0} \approx 0.107
\end{align*}
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<div class="para">So the power of the test is <span class="process-math">\(0.268 + 0.107 = 0.375\text{.}\)</span>
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<div class="autopermalink" data-description="Solution 1.c.1"><a href="#quiz-04-3-1-4-2" title="Copy heading and permalink for Solution 1.c.1" aria-label="Copy heading and permalink for Solution 1.c.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 1.c"><a href="#quiz-04-3-1-4" title="Copy heading and permalink for Task 1.c" aria-label="Copy heading and permalink for Task 1.c">🔗</a></div></article><div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-04-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><div class="autopermalink" data-description="Worksheet: Quiz 4"><a href="#quiz-04" title="Copy heading and permalink for Worksheet: Quiz 4" aria-label="Copy heading and permalink for Worksheet: Quiz 4">🔗</a></div></section>
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