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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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<div class="para" id="Final-Exam-Review-3-2">You will be allowed to use a scientific calculator (<em class="emphasis">not</em> a graphing calculator, <em class="emphasis">not</em> a calculator app on your phone). You may not share a calculator with another student; you must use your own calculator.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-3-2" 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="Subsection: Allowed Materials"><a href="#Final-Exam-Review-3" title="Copy heading and permalink for Subsection: Allowed Materials" aria-label="Copy heading and permalink for Subsection: Allowed Materials">🔗</a></div></section><section class="exercises" id="Final-Exam-Review-4"><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="Final-Exam-Review-4-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-1-1"><div class="para" id="Final-Exam-Review-4-1-1-1">Consider the sets <span class="process-math">\(A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\text{,}\)</span> <span class="process-math">\(B = \{2, 4, 9, 10, 12, 14, 19\}\text{,}\)</span> and <span class="process-math">\(C = \{9, 10, 11, 14, 16, 17, 20\}\text{,}\)</span> which are all subsets of <span class="process-math">\(\Omega = \{1, 2, 3, \dotsc, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-1-1" 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="Final-Exam-Review-4-1-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-1-2-1-1">Find <span class="process-math">\(A - (B \cap C)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-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="solutions"><details id="Final-Exam-Review-4-1-2-2" 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="Final-Exam-Review-4-1-2-2-1">
<span class="process-math">\(\{1, 2, 3, 4, 5, 6, 7, 8\}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.a.1"><a href="#Final-Exam-Review-4-1-2-2" title="Copy heading and permalink for Answer 1.a.1" aria-label="Copy heading and permalink for Answer 1.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.a"><a href="#Final-Exam-Review-4-1-2" 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="Final-Exam-Review-4-1-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-1-3-1-1">Find <span class="process-math">\(|A|\text{,}\)</span> <span class="process-math">\(|B|\text{,}\)</span> <span class="process-math">\(|C|\text{,}\)</span> <span class="process-math">\(|A\cup B|\text{,}\)</span> <span class="process-math">\(|A \cap B|\text{,}\)</span> <span class="process-math">\(|B\cap C|\text{,}\)</span> <span class="process-math">\(|A\cap C|\text{,}\)</span> and <span class="process-math">\(|A\cup B\cup C|\text{.}\)</span> Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-1-3-2" 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="Final-Exam-Review-4-1-3-2-1">
<span class="process-math">\(|A| = 10\text{,}\)</span> <span class="process-math">\(|B| = 7\text{,}\)</span> <span class="process-math">\(|C| = 7\text{,}\)</span> <span class="process-math">\(|A \cup B| = 13\text{,}\)</span> <span class="process-math">\(|A \cap B| = 4\text{,}\)</span> <span class="process-math">\(|B\cap C| = 3\text{,}\)</span> <span class="process-math">\(|A\cap C| = 2\text{,}\)</span> <span class="process-math">\(|A\cup B\cup C| = 17\text{.}\)</span> In particular, note that <span class="process-math">\(|A\cup B| = 13 \neq 10 + 7 = |A| + |B|\text{,}\)</span> so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.b.1"><a href="#Final-Exam-Review-4-1-3-2" title="Copy heading and permalink for Answer 1.b.1" aria-label="Copy heading and permalink for Answer 1.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.b"><a href="#Final-Exam-Review-4-1-3" 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="Final-Exam-Review-4-1-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-1-4-1-1">Find <span class="process-math">\(A^c\)</span> and <span class="process-math">\((A\cup B)^c\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-1-4-2" 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="Final-Exam-Review-4-1-4-2-1">
<span class="process-math">\(A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}\text{,}\)</span> <span class="process-math">\((A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-1-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 1.c.1"><a href="#Final-Exam-Review-4-1-4-2" title="Copy heading and permalink for Answer 1.c.1" aria-label="Copy heading and permalink for Answer 1.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 1.c"><a href="#Final-Exam-Review-4-1-4" 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="Exercise 1"><a href="#Final-Exam-Review-4-1" title="Copy heading and permalink for Exercise 1" aria-label="Copy heading and permalink for Exercise 1">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-2-1-1">Suppose we have a 6-sided die that’s weighted to roll a 6 half of the time. We roll the die two times. List the set of all possible results. [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result <span class="process-math">\((4, 2)\)</span>---rolling a 4 and then a 2.]<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-2-2" 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 logical" id="Final-Exam-Review-4-2-2-1">
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-sample-space.html" id="Final-Exam-Review-4-2-2-1-1">
\begin{align*}
\Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), \\
\amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), \\
\amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), \\
\amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\}
\end{align*}
</div>
<div class="para">Note that <span class="process-math">\(\Omega\)</span> simply lists outcomes with no reference to the probabilities. So the answer here is the same as in <a href="notes-01-13.html#example-sample-space" class="xref" data-knowl="./knowl/xref/example-sample-space.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 4">Example 4</a>.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 2.1"><a href="#Final-Exam-Review-4-2-2" title="Copy heading and permalink for Answer 2.1" aria-label="Copy heading and permalink for Answer 2.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 2"><a href="#Final-Exam-Review-4-2" title="Copy heading and permalink for Exercise 2" aria-label="Copy heading and permalink for Exercise 2">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-3-1-1">Suppose we flip a coin two times. List the set of all possible results. What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-3-2" 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="Final-Exam-Review-4-3-2-1">For two flips: <span class="process-math">\(\Omega = \{ HH, HT, TH, TT \}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Final-Exam-Review-4-3-2-2">For three flips: <span class="process-math">\(\Omega = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-3-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para logical" id="Final-Exam-Review-4-3-2-3">
<div class="para">For four flips:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-3-2-3-1">
\begin{align*}
\Omega = \{ \amp HHHH, HHHT, HHTH, HTHH, \\
\amp THHH, HHTT, HTHT, HTTH, \\
\amp THHT, THTH, TTHH, HTTT, \\
\amp THTT, TTHT, TTTH, TTTT \}.
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-3-2-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Final-Exam-Review-4-3-2-4">Each additional flip doubles the number of outcomes. So, with ten flips, we’ll have <span class="process-math">\(|\Omega| = 2^{10} = 1024\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-3-2-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 3.1"><a href="#Final-Exam-Review-4-3-2" title="Copy heading and permalink for Answer 3.1" aria-label="Copy heading and permalink for Answer 3.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 3"><a href="#Final-Exam-Review-4-3" title="Copy heading and permalink for Exercise 3" aria-label="Copy heading and permalink for Exercise 3">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-4"><h4 class="heading"><span class="codenumber">4<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-4-1-1">If we roll a 6-sided die ten times, how many possible results will there be?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-4-2" 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="Final-Exam-Review-4-4-2-1">Each additional roll will multiply the number of outcomes by 6. So, with 10 rolls, we’ll have <span class="process-math">\(|\Omega| = 6^{10}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 4.1"><a href="#Final-Exam-Review-4-4-2" title="Copy heading and permalink for Answer 4.1" aria-label="Copy heading and permalink for Answer 4.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 4"><a href="#Final-Exam-Review-4-4" title="Copy heading and permalink for Exercise 4" aria-label="Copy heading and permalink for Exercise 4">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-5"><h4 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-5-1-1">Consider the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6, 7, 8\}\)</span> with probability distribution below. Calculate the probabilities of <span class="process-math">\(A = \{1, 3, 7, 8\}\text{,}\)</span> <span class="process-math">\(B = \{2, 3, 6, 7\}\text{,}\)</span> <span class="process-math">\(A\cup B\text{,}\)</span> and <span class="process-math">\(A \cap B\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-5-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">144<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 144: "><a href="#Final-Exam-Review-4-5-1-2" title="Copy heading and permalink for Table 144: " aria-label="Copy heading and permalink for Table 144: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="solutions"><details id="Final-Exam-Review-4-5-2" 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="Final-Exam-Review-4-5-2-1">
<span class="process-math">\(\Pr(A) = 0.45, \Pr(B) = 0.4, \Pr(A \cup B) = 0.6, \Pr(A \cap B) = 0.25.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-5-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 5.1"><a href="#Final-Exam-Review-4-5-2" title="Copy heading and permalink for Answer 5.1" aria-label="Copy heading and permalink for Answer 5.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 5"><a href="#Final-Exam-Review-4-5" title="Copy heading and permalink for Exercise 5" aria-label="Copy heading and permalink for Exercise 5">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-6"><h4 class="heading"><span class="codenumber">6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-6-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, the probabilities scale by the same amount as the face values. For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-6-2" 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">
<figure class="table table-like" id="Final-Exam-Review-4-6-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">145<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for a Linearly Scaled Die<div class="autopermalink" data-description="Table 145: Probability Distribution for a Linearly Scaled Die"><a href="#Final-Exam-Review-4-6-2-1" title="Copy heading and permalink for Table 145: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 145: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/21\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/21\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(3/21\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(4/21\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(5/21\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(6/21\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 6.1"><a href="#Final-Exam-Review-4-6-2" title="Copy heading and permalink for Answer 6.1" aria-label="Copy heading and permalink for Answer 6.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 6"><a href="#Final-Exam-Review-4-6" title="Copy heading and permalink for Exercise 6" aria-label="Copy heading and permalink for Exercise 6">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-7"><h4 class="heading"><span class="codenumber">7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-7-1-1">Suppose a die has the values <span class="process-math">\(1, 2, 3, 4, 5, 6\)</span> on the faces, but the die is not fair. Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. Write a probability distribution table for this die.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-7-2" 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">
<figure class="table table-like" id="Final-Exam-Review-4-7-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">146<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Probability Distribution for an Even-biased Die<div class="autopermalink" data-description="Table 146: Probability Distribution for an Even-biased Die"><a href="#Final-Exam-Review-4-7-2-1" title="Copy heading and permalink for Table 146: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 146: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/9\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/9\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/9\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/9\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(1/9\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(2/9\)</span></td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 7.1"><a href="#Final-Exam-Review-4-7-2" title="Copy heading and permalink for Answer 7.1" aria-label="Copy heading and permalink for Answer 7.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 7"><a href="#Final-Exam-Review-4-7" title="Copy heading and permalink for Exercise 7" aria-label="Copy heading and permalink for Exercise 7">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-8"><h4 class="heading"><span class="codenumber">8<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-8-1-1">A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. For each value of <span class="process-math">\(n = 1, 2, 3, \dotsc\text{,}\)</span> find the probability of the toxin molecule leaving the cell during the <span class="process-math">\(n\)</span>th minute. What is the probability of the molecule leaving the cell during the first 3 minutes?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-8-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-8-2" 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="Final-Exam-Review-4-8-2-1">For short, write <span class="process-math">\(\Pr(n)\)</span> to mean the probability of the toxin molecule leaving during the <span class="process-math">\(n\)</span>th minute. Then <span class="process-math">\(\Pr(n) = (0.7)^{n - 1} (0.3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-8-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="Final-Exam-Review-4-8-2-2">The probability of leaving during the first 3 minutes is <span class="process-math">\(\Pr(1) + \Pr(2) + \Pr(3) = 0.657.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-8-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 8.1"><a href="#Final-Exam-Review-4-8-2" title="Copy heading and permalink for Answer 8.1" aria-label="Copy heading and permalink for Answer 8.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 8"><a href="#Final-Exam-Review-4-8" title="Copy heading and permalink for Exercise 8" aria-label="Copy heading and permalink for Exercise 8">🔗</a></div></article><div class="exercisegroup" id="Final-Exam-Review-4-9">
<h4 class="heading"><span class="title">Exercise Group.</span></h4>
<div class="introduction" id="Final-Exam-Review-4-9-1"><div class="para" id="Final-Exam-Review-4-9-1-1">In each of the following scenarios with given events <span class="process-math">\(A\)</span> and <span class="process-math">\(B\text{,}\)</span> alculate <span class="process-math">\(\Pr(A), \Pr(B)\text{,}\)</span> <span class="process-math">\(\Pr(A\cap B)\text{,}\)</span> <span class="process-math">\(\Pr(A \mid B)\text{,}\)</span> and <span class="process-math">\(\Pr(B \mid A)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-9-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></div>
<div class="exercisegroup-exercises">
<article class="exercise exercise-like" id="Final-Exam-Review-4-9-2"><h5 class="heading"><span class="codenumber">9<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
<div class="para" id="Final-Exam-Review-4-9-2-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-9-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-9-2-2" 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 logical" id="Final-Exam-Review-4-9-2-2-1">
<div class="displaymath process-math" id="Final-Exam-Review-4-9-2-2-1-1">
\begin{align*}
A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 4), (3, 5), (3, 6), \\
\amp (4, 5), (4, 6), \\
\amp (5, 6)\} \\
A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-9-2-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{6/36}{18/36} = \frac{6}{18} = \frac{1}{3}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-9-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 9.1"><a href="#Final-Exam-Review-4-9-2-2" title="Copy heading and permalink for Answer 9.1" aria-label="Copy heading and permalink for Answer 9.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 9"><a href="#Final-Exam-Review-4-9-2" title="Copy heading and permalink for Exercise 9" aria-label="Copy heading and permalink for Exercise 9">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-9-3"><h5 class="heading"><span class="codenumber">10<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h5>
<div class="para" id="Final-Exam-Review-4-9-3-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-9-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-9-3-2" 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 logical" id="Final-Exam-Review-4-9-3-2-1">
<div class="displaymath process-math" id="Final-Exam-Review-4-9-3-2-1-1">
\begin{align*}
A \amp = \{ HHH, HHT, TTH, TTT \} \\
B \amp = \{ HHH, HHT, HTH, THH \} \\
A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-9-3-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} \\
\Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2}
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-9-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 10.1"><a href="#Final-Exam-Review-4-9-3-2" title="Copy heading and permalink for Answer 10.1" aria-label="Copy heading and permalink for Answer 10.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 10"><a href="#Final-Exam-Review-4-9-3" title="Copy heading and permalink for Exercise 10" aria-label="Copy heading and permalink for Exercise 10">🔗</a></div></article>
</div>
<div class="autopermalink" data-description="Exercise Group 9–10"><a href="#Final-Exam-Review-4-9" title="Copy heading and permalink for Exercise Group 9–10" aria-label="Copy heading and permalink for Exercise Group 9–10">🔗</a></div>
</div>
<article class="exercise exercise-like" id="Final-Exam-Review-4-10"><h4 class="heading"><span class="codenumber">11<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-10-1"><div class="para" id="Final-Exam-Review-4-10-1-1">A diagnostic test is developed to detect a disease present in 3.2% of the population. For a patient who has the disease, the test will accurately give a positive result 65% of the time. When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-10-1-1" 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="Final-Exam-Review-4-10-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-10-2-1-1">For a patient who receives a positive test, what is the probability they have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-10-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-10-2-2" 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 logical" id="Final-Exam-Review-4-10-2-2-1">
<div class="para">Let <span class="process-math">\(P\)</span> be the event of testing positive and <span class="process-math">\(D\)</span> the event of having the disease. Then the prevalence <span class="process-math">\(\Pr(D)\)</span> is given as 3.2%, or 0.032. The sensitivity is <span class="process-math">\(\Pr(P\mid D) = 0.65\text{,}\)</span> and the specificity is <span class="process-math">\(\Pr(P^c\mid D^c) = 0.999\text{.}\)</span> So, according to Bayes’ Theorem:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-10-2-2-1-6">
\begin{align*}
\Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\mid D^c))\Pr(D^c)} \\
\amp = \frac{(0.65)(0.032)}{(0.65)(0.032) + (1 - 0.999)(1 - 0.032)} \\
\amp \approx 0.96
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-10-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 11.a.1"><a href="#Final-Exam-Review-4-10-2-2" title="Copy heading and permalink for Answer 11.a.1" aria-label="Copy heading and permalink for Answer 11.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 11.a"><a href="#Final-Exam-Review-4-10-2" title="Copy heading and permalink for Task 11.a" aria-label="Copy heading and permalink for Task 11.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-10-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-10-3-1-1">For a patient who receives a negative test, what is the probability they do not have the disease?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-10-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-10-3-2" 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 logical" id="Final-Exam-Review-4-10-3-2-1">
<div class="displaymath process-math" id="Final-Exam-Review-4-10-3-2-1-1">
\begin{align*}
\Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + (1 - \Pr(P\mid D))\Pr(D)} \\
\amp = \frac{(0.999)(1 - 0.032)}{(0.999)(1 - 0.032) + (1 - 0.65)(0.032)} \\
\amp \approx 0.99
\end{align*}
</div>
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<div class="autopermalink" data-description="Answer 11.b.1"><a href="#Final-Exam-Review-4-10-3-2" title="Copy heading and permalink for Answer 11.b.1" aria-label="Copy heading and permalink for Answer 11.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 11.b"><a href="#Final-Exam-Review-4-10-3" title="Copy heading and permalink for Task 11.b" aria-label="Copy heading and permalink for Task 11.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 11"><a href="#Final-Exam-Review-4-10" title="Copy heading and permalink for Exercise 11" aria-label="Copy heading and permalink for Exercise 11">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-11"><h4 class="heading"><span class="codenumber">12<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-11-1-1">An experiment consists of rolling a fair die two times. Let <span class="process-math">\(A\)</span> be the event that the sum is even, and let <span class="process-math">\(B\)</span> be the event that the second roll is higher than the first. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-11-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-11-2" 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 logical" id="Final-Exam-Review-4-11-2-1">
<div class="displaymath process-math" id="Final-Exam-Review-4-11-2-1-1">
\begin{align*}
A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), \\
\amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), \\
\amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} \\
B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), \\
\amp (2, 3), (2, 4), (2, 5), (2, 6), \\
\amp (3, 4), (3, 5), (3, 6), \\
\amp (4, 5), (4, 6), \\
\amp (5, 6)\} \\
A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{18}{36} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{15}{36} = \frac{5}{12}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-11-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \neq \Pr(A),
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are not independent.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-11-2-1" 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 12.1"><a href="#Final-Exam-Review-4-11-2" title="Copy heading and permalink for Answer 12.1" aria-label="Copy heading and permalink for Answer 12.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 12"><a href="#Final-Exam-Review-4-11" title="Copy heading and permalink for Exercise 12" aria-label="Copy heading and permalink for Exercise 12">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-12"><h4 class="heading"><span class="codenumber">13<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-12-1-1">An experiment consists of flipping a fair coin three times. Let <span class="process-math">\(A\)</span> be the event that the first and second flips match. Let <span class="process-math">\(B\)</span> be the event that there are at least two heads. Are <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-12-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-12-2" 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 logical" id="Final-Exam-Review-4-12-2-1">
<div class="displaymath process-math" id="Final-Exam-Review-4-12-2-1-1">
\begin{align*}
A \amp = \{ HHH, HHT, TTH, TTT \} \\
B \amp = \{ HHH, HHT, HTH, THH \} \\
A\cap B \amp = \{HHH, HHT\}
\end{align*}
</div>
<div class="para">So <span class="process-math">\(\Pr(A) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> <span class="process-math">\(\Pr(B) = \frac{4}{8} = \frac{1}{2}\text{,}\)</span> and <span class="process-math">\(\Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}\text{.}\)</span> Finally:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-12-2-1-5">
\begin{align*}
\Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} = \Pr(A),
\end{align*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<div class="autopermalink" data-description="Answer 13.1"><a href="#Final-Exam-Review-4-12-2" title="Copy heading and permalink for Answer 13.1" aria-label="Copy heading and permalink for Answer 13.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 13"><a href="#Final-Exam-Review-4-12" title="Copy heading and permalink for Exercise 13" aria-label="Copy heading and permalink for Exercise 13">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-13"><h4 class="heading"><span class="codenumber">14<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-13-1-1">Let <span class="process-math">\(A = \{1, 2, 3\}\)</span> and <span class="process-math">\(B = \{3, 4, 5\}\)</span> be events in the sample space <span class="process-math">\(\Omega = \{1, 2, 3, 4, 5, 6\}\text{.}\)</span> Create a probability distribution for <span class="process-math">\(\Omega\)</span> so that <span class="process-math">\(A, B\)</span> are independent.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-13-2" 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">
<figure class="table table-like" id="Final-Exam-Review-4-13-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">147<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Example Distribution<div class="autopermalink" data-description="Table 147: Example Distribution"><a href="#Final-Exam-Review-4-13-2-1" title="Copy heading and permalink for Table 147: Example Distribution" aria-label="Copy heading and permalink for Table 147: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure><div class="para logical" id="Final-Exam-Review-4-13-2-2">
<div class="para">Now <span class="process-math">\(\Pr(A) = 0.5\text{,}\)</span> <span class="process-math">\(\Pr(B) = 0.4\text{,}\)</span> and</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-13-2-2-3">
\begin{gather*}
\Pr(A\cap B) = 0.2 = (0.5)(0.4) = \Pr(A)\Pr(B),
\end{gather*}
</div>
<div class="para">so <span class="process-math">\(A\)</span> and <span class="process-math">\(B\)</span> are independent.</div>
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<div class="autopermalink" data-description="Answer 14.1"><a href="#Final-Exam-Review-4-13-2" title="Copy heading and permalink for Answer 14.1" aria-label="Copy heading and permalink for Answer 14.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 14"><a href="#Final-Exam-Review-4-13" title="Copy heading and permalink for Exercise 14" aria-label="Copy heading and permalink for Exercise 14">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-14"><h4 class="heading"><span class="codenumber">15<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-14-1-1">An experiment consists of flipping a biased coin 20 times. If the coin comes up heads with probability <span class="process-math">\(p = 0.3\text{,}\)</span> find the probability of seeing 5 heads. Find the probability of seeing up to (and including) 3 heads.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-14-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-14-2" 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="Final-Exam-Review-4-14-2-1">
<div class="para">Let <span class="process-math">\(S\)</span> be the number of heads. Then <span class="process-math">\(S \sim \Bin(20, 0.3)\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-14-2-1-3">
\begin{align*}
\Pr(S = 5) \amp = {20 \choose 5} (0.3)^5 (0.7)^{20 - 5} \\
\amp = \frac{20!}{(5!)(15!)} (0.3)^5 (0.7)^{15} \\
\amp = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} (0.3)^5 (0.7)^{15} \\
\amp = (19 \times 3 \times 17 \times 16) (0.3)^5 (0.7)^{15} \\
\amp \approx 0.179
\end{align*}
</div>
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<div class="autopermalink" data-description="Solution 15.1"><a href="#Final-Exam-Review-4-14-2" title="Copy heading and permalink for Solution 15.1" aria-label="Copy heading and permalink for Solution 15.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 15"><a href="#Final-Exam-Review-4-14" title="Copy heading and permalink for Exercise 15" aria-label="Copy heading and permalink for Exercise 15">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-15"><h4 class="heading"><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-15-1"><div class="para" id="Final-Exam-Review-4-15-1-1">An experiment consists of flipping a coin repeatedly until we first see heads.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-15-1-1" 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="Final-Exam-Review-4-15-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-15-2-1-1">If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-15-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-15-2-2" 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="Final-Exam-Review-4-15-2-2-1">
<div class="para">Let <span class="process-math">\(T\)</span> be the number of flips until we see heads. Then <span class="process-math">\(T\)</span> is geometric with parameter <span class="process-math">\(p = 0.4\text{,}\)</span> so:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-15-2-2-1-4">
\begin{align*}
\Pr(T = k) \amp = (1-0.4)^{k-1}(0.4) = 0.6^{k-1} \cdot 0.4
\end{align*}
</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-15-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Solution 16.a.1"><a href="#Final-Exam-Review-4-15-2-2" title="Copy heading and permalink for Solution 16.a.1" aria-label="Copy heading and permalink for Solution 16.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 16.a"><a href="#Final-Exam-Review-4-15-2" title="Copy heading and permalink for Task 16.a" aria-label="Copy heading and permalink for Task 16.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-15-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-15-3-1-1">Which flip has the highest chance of being the first flip to come up heads?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-15-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 16.b"><a href="#Final-Exam-Review-4-15-3" title="Copy heading and permalink for Task 16.b" aria-label="Copy heading and permalink for Task 16.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 16"><a href="#Final-Exam-Review-4-15" title="Copy heading and permalink for Exercise 16" aria-label="Copy heading and permalink for Exercise 16">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-16"><h4 class="heading"><span class="codenumber">17<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-16-1-1">A particular store has an average of 20 customers each hour. During a 4-hour afternoon shift, what is the probability of serving 80 customers.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-16-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 17"><a href="#Final-Exam-Review-4-16" title="Copy heading and permalink for Exercise 17" aria-label="Copy heading and permalink for Exercise 17">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-17"><h4 class="heading"><span class="codenumber">18<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-17-1"><div class="para" id="Final-Exam-Review-4-17-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = k(x - \sqrt{x})\)</span> for some constant <span class="process-math">\(k\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-17-1-1" 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="Final-Exam-Review-4-17-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-17-2-1-1">What is the value of <span class="process-math">\(k\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-17-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-17-2-2" 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="Final-Exam-Review-4-17-2-2-1">
<span class="process-math">\(k = \frac{6}{17}.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-17-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 18.a.1"><a href="#Final-Exam-Review-4-17-2-2" title="Copy heading and permalink for Answer 18.a.1" aria-label="Copy heading and permalink for Answer 18.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 18.a"><a href="#Final-Exam-Review-4-17-2" title="Copy heading and permalink for Task 18.a" aria-label="Copy heading and permalink for Task 18.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-17-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-17-3-1-1">Find <span class="process-math">\(\Pr(2 \leq X \leq 3).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-17-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-17-3-2" 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="Final-Exam-Review-4-17-3-2-1">
<span class="process-math">\(\Pr(2 \leq X \leq 3) \approx 0.325\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-17-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 18.b.1"><a href="#Final-Exam-Review-4-17-3-2" title="Copy heading and permalink for Answer 18.b.1" aria-label="Copy heading and permalink for Answer 18.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 18.b"><a href="#Final-Exam-Review-4-17-3" title="Copy heading and permalink for Task 18.b" aria-label="Copy heading and permalink for Task 18.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 18"><a href="#Final-Exam-Review-4-17" title="Copy heading and permalink for Exercise 18" aria-label="Copy heading and permalink for Exercise 18">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-18"><h4 class="heading"><span class="codenumber">19<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-18-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find the c.d.f. <span class="process-math">\(F(x)\text{.}\)</span> Use your c.d.f. to find <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-18-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-18-2" 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="Final-Exam-Review-4-18-2-1">
<span class="process-math">\(\frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}\text{.}\)</span> <span class="process-math">\(\Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-18-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 19.1"><a href="#Final-Exam-Review-4-18-2" title="Copy heading and permalink for Answer 19.1" aria-label="Copy heading and permalink for Answer 19.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 19"><a href="#Final-Exam-Review-4-18" title="Copy heading and permalink for Exercise 19" aria-label="Copy heading and permalink for Exercise 19">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-19"><h4 class="heading"><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-19-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([2, 3]\)</span> has c.d.f. <span class="process-math">\(F(x) = \frac{x^3}{3} - x^2 + 4\text{.}\)</span> Find the p.d.f. <span class="process-math">\(f(x)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-19-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-19-2" 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="Final-Exam-Review-4-19-2-1">
<span class="process-math">\(f(x) = x^2 - 2x.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-19-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 20.1"><a href="#Final-Exam-Review-4-19-2" title="Copy heading and permalink for Answer 20.1" aria-label="Copy heading and permalink for Answer 20.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 20"><a href="#Final-Exam-Review-4-19" title="Copy heading and permalink for Exercise 20" aria-label="Copy heading and permalink for Exercise 20">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-20"><h4 class="heading"><span class="codenumber">21<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-20-1-1">Consider <span class="process-math">\(X, Y\)</span> with the joint distribution table below. Are <span class="process-math">\(X, Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-20-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">148<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint distribution for <span class="process-math">\(X, Y\)</span><div class="autopermalink" data-description="Table 148: Joint distribution for X, Y"><a href="#Final-Exam-Review-4-20-1-2" title="Copy heading and permalink for Table 148: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 148: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.2</td>
<td class="c m b1 r0 l0 t0 lines">0.3</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.4</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="solutions"><details id="Final-Exam-Review-4-20-2" 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="Final-Exam-Review-4-20-2-1">No. For example, <span class="process-math">\(\Pr(X = 0, Y = 0) \neq \Pr(X = 0)\Pr(Y = 0).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-20-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 21.1"><a href="#Final-Exam-Review-4-20-2" title="Copy heading and permalink for Answer 21.1" aria-label="Copy heading and permalink for Answer 21.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 21"><a href="#Final-Exam-Review-4-20" title="Copy heading and permalink for Exercise 21" aria-label="Copy heading and permalink for Exercise 21">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-21"><h4 class="heading"><span class="codenumber">22<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para logical" id="Final-Exam-Review-4-21-1-1">
<div class="para">Suppose <span class="process-math">\(X, Y\)</span> have the distributions:</div>
<div class="displaymath process-math" id="Final-Exam-Review-4-21-1-1-2">
\begin{align*}
\Pr(X = 0) \amp = 0.1 \amp \Pr(Y = 0) \amp = 0.4 \\
\Pr(X = 1) \amp = 0.4 \amp \Pr(Y = 1) \amp = 0.6 \\
\Pr(X = 2) \amp = 0.5
\end{align*}
</div>
<div class="para">Assuming <span class="process-math">\(X, Y\)</span> are independent, write a joint distribution table.</div>
<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-21-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-21-2" 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">
<figure class="table table-like" id="Final-Exam-Review-4-21-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">149<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 149: Joint Distribution"><a href="#Final-Exam-Review-4-21-2-1" title="Copy heading and permalink for Table 149: Joint Distribution" aria-label="Copy heading and permalink for Table 149: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="c m b1 r1 l0 t0 lines"></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></td>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.04</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.06</td>
<td class="c m b0 r1 l0 t0 lines">0.24</td>
<td class="c m b0 r0 l0 t0 lines">0.3</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Answer 22.1"><a href="#Final-Exam-Review-4-21-2" title="Copy heading and permalink for Answer 22.1" aria-label="Copy heading and permalink for Answer 22.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 22"><a href="#Final-Exam-Review-4-21" title="Copy heading and permalink for Exercise 22" aria-label="Copy heading and permalink for Exercise 22">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-22"><h4 class="heading"><span class="codenumber">23<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-22-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-22-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-22-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">150<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 150: "><a href="#Final-Exam-Review-4-22-1-2" title="Copy heading and permalink for Table 150: " aria-label="Copy heading and permalink for Table 150: ">🔗</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"><span class="process-math">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="solutions"><details id="Final-Exam-Review-4-22-2" 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="Final-Exam-Review-4-22-2-1">4.8.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-22-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 23.1"><a href="#Final-Exam-Review-4-22-2" title="Copy heading and permalink for Answer 23.1" aria-label="Copy heading and permalink for Answer 23.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 23"><a href="#Final-Exam-Review-4-22" title="Copy heading and permalink for Exercise 23" aria-label="Copy heading and permalink for Exercise 23">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-23"><h4 class="heading"><span class="codenumber">24<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-23-1"><div class="para" id="Final-Exam-Review-4-23-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-1-1" 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="Final-Exam-Review-4-23-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-23-2-1-1">If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\E(N)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-23-2-2" 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="Final-Exam-Review-4-23-2-2-1">40.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-2-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.a.1"><a href="#Final-Exam-Review-4-23-2-2" title="Copy heading and permalink for Answer 24.a.1" aria-label="Copy heading and permalink for Answer 24.a.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.a"><a href="#Final-Exam-Review-4-23-2" title="Copy heading and permalink for Task 24.a" aria-label="Copy heading and permalink for Task 24.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-23-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-23-3-1-1">What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-23-3-2" 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="Final-Exam-Review-4-23-3-2-1">48.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.b.1"><a href="#Final-Exam-Review-4-23-3-2" title="Copy heading and permalink for Answer 24.b.1" aria-label="Copy heading and permalink for Answer 24.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.b"><a href="#Final-Exam-Review-4-23-3" title="Copy heading and permalink for Task 24.b" aria-label="Copy heading and permalink for Task 24.b">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-23-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-23-4-1-1">What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-23-4-2" 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="Final-Exam-Review-4-23-4-2-1">100.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-23-4-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 24.c.1"><a href="#Final-Exam-Review-4-23-4-2" title="Copy heading and permalink for Answer 24.c.1" aria-label="Copy heading and permalink for Answer 24.c.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 24.c"><a href="#Final-Exam-Review-4-23-4" title="Copy heading and permalink for Task 24.c" aria-label="Copy heading and permalink for Task 24.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 24"><a href="#Final-Exam-Review-4-23" title="Copy heading and permalink for Exercise 24" aria-label="Copy heading and permalink for Exercise 24">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-24"><h4 class="heading"><span class="codenumber">25<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-24-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\E(Y) = -2\text{,}\)</span> and <span class="process-math">\(\E(Z) = 1\text{,}\)</span> what is <span class="process-math">\(\E(4X + 5Y - Z + 3)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-24-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-24-2" 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="Final-Exam-Review-4-24-2-1">4.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-24-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 25.1"><a href="#Final-Exam-Review-4-24-2" title="Copy heading and permalink for Answer 25.1" aria-label="Copy heading and permalink for Answer 25.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 25"><a href="#Final-Exam-Review-4-24" title="Copy heading and permalink for Exercise 25" aria-label="Copy heading and permalink for Exercise 25">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-25"><h4 class="heading"><span class="codenumber">26<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-25-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-25-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-25-2" 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="Final-Exam-Review-4-25-2-1">2/3.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-25-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 26.1"><a href="#Final-Exam-Review-4-25-2" title="Copy heading and permalink for Answer 26.1" aria-label="Copy heading and permalink for Answer 26.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 26"><a href="#Final-Exam-Review-4-25" title="Copy heading and permalink for Exercise 26" aria-label="Copy heading and permalink for Exercise 26">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-26"><h4 class="heading"><span class="codenumber">27<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-26-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\E(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-26-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-26-2" 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="Final-Exam-Review-4-26-2-1">
<span class="process-math">\(\frac{4}{3}\ln(4) \approx 1.85.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-26-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 27.1"><a href="#Final-Exam-Review-4-26-2" title="Copy heading and permalink for Answer 27.1" aria-label="Copy heading and permalink for Answer 27.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 27"><a href="#Final-Exam-Review-4-26" title="Copy heading and permalink for Exercise 27" aria-label="Copy heading and permalink for Exercise 27">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-27"><h4 class="heading"><span class="codenumber">28<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-27-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\E(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-27-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-27-2" 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="Final-Exam-Review-4-27-2-1">
<span class="process-math">\(\frac{1}{2}\left( \ln(2) + \frac{7}{3}\right) \approx 1.513.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-27-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 28.1"><a href="#Final-Exam-Review-4-27-2" title="Copy heading and permalink for Answer 28.1" aria-label="Copy heading and permalink for Answer 28.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 28"><a href="#Final-Exam-Review-4-27" title="Copy heading and permalink for Exercise 28" aria-label="Copy heading and permalink for Exercise 28">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-28"><h4 class="heading"><span class="codenumber">29<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-28-1-1">Consider a random variable <span class="process-math">\(X\)</span> with probability distribution below. Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-28-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-28-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">151<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 151: "><a href="#Final-Exam-Review-4-28-1-2" title="Copy heading and permalink for Table 151: " aria-label="Copy heading and permalink for Table 151: ">🔗</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"><span class="process-math">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">1</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">2</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">0.2</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">4</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">5</td>
<td class="c m b0 r0 l0 t0 lines">0.15</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">6</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">7</td>
<td class="c m b0 r0 l0 t0 lines">0.05</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">8</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">9</td>
<td class="c m b0 r0 l0 t0 lines">0.1</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 29"><a href="#Final-Exam-Review-4-28" title="Copy heading and permalink for Exercise 29" aria-label="Copy heading and permalink for Exercise 29">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-29"><h4 class="heading"><span class="codenumber">30<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-29-1-1">Suppose we flip a coin <span class="process-math">\(n = 100\)</span> times, and let <span class="process-math">\(N\)</span> count the number of heads. If the coin comes up heads on a flip with probability <span class="process-math">\(p = 0.4\text{,}\)</span> what is <span class="process-math">\(\Var(N)\text{?}\)</span> What if <span class="process-math">\(n = 80\)</span> and <span class="process-math">\(p = 0.6\text{?}\)</span> What if <span class="process-math">\(n = 200\)</span> and <span class="process-math">\(p = 0.5\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-29-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 30"><a href="#Final-Exam-Review-4-29" title="Copy heading and permalink for Exercise 30" aria-label="Copy heading and permalink for Exercise 30">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-30"><h4 class="heading"><span class="codenumber">31<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-30-1-1">If <span class="process-math">\(\E(X) = 3\text{,}\)</span> <span class="process-math">\(\Var(X) = 2\text{,}\)</span> what is <span class="process-math">\(\E(X^2)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-30-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 31"><a href="#Final-Exam-Review-4-30" title="Copy heading and permalink for Exercise 31" aria-label="Copy heading and permalink for Exercise 31">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-31"><h4 class="heading"><span class="codenumber">32<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-31-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([0, 1]\)</span> has p.d.f. <span class="process-math">\(f(x) = 2x\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-31-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 32"><a href="#Final-Exam-Review-4-31" title="Copy heading and permalink for Exercise 32" aria-label="Copy heading and permalink for Exercise 32">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-32"><h4 class="heading"><span class="codenumber">33<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-32-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 4]\)</span> has p.d.f. <span class="process-math">\(f(x) = \frac{4}{3x^2}\text{.}\)</span> What is <span class="process-math">\(\Var(X)\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-32-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 33"><a href="#Final-Exam-Review-4-32" title="Copy heading and permalink for Exercise 33" aria-label="Copy heading and permalink for Exercise 33">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-33"><h4 class="heading"><span class="codenumber">34<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-33-1-1">A continuous random variable <span class="process-math">\(X\)</span> taking values in <span class="process-math">\([1, 2]\)</span> has p.d.f. <span class="process-math">\(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\)</span> Find <span class="process-math">\(\Var(X)\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-33-1-1" 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="Exercise 34"><a href="#Final-Exam-Review-4-33" title="Copy heading and permalink for Exercise 34" aria-label="Copy heading and permalink for Exercise 34">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-34"><h4 class="heading"><span class="codenumber">35<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-34-1-1">Suppose a coin has an unknown probability of coming up heads. We perform the experiment in <span class="process-math">\(n\)</span> independent trials, during which it takes <span class="process-math">\(k_1, k_2, \dotsc, k_n\)</span> flips to see our first heads in each trial. Find a "common sense" MLE formula for the geometric distribution.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-34-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 35"><a href="#Final-Exam-Review-4-34" title="Copy heading and permalink for Exercise 35" aria-label="Copy heading and permalink for Exercise 35">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-35"><h4 class="heading"><span class="codenumber">36<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-35-1-1">A particular store owner wants to approximate the average hourly rate at which customers come into the store. They observe 80 customers enter during a particular 4-hour shift. What is the maximum likelihood estimation for the hourly customer rate?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-35-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 36"><a href="#Final-Exam-Review-4-35" title="Copy heading and permalink for Exercise 36" aria-label="Copy heading and permalink for Exercise 36">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-36"><h4 class="heading"><span class="codenumber">37<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-36-1-1">A radioactive material emits particles at an unknown probabilistic rate <span class="process-math">\(\lambda\)</span> particles per minute. We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. Write the likelihood function <span class="process-math">\(\mathcal{L}(\lambda)\)</span> based on this data. What is the maximum likelihood estimation for <span class="process-math">\(\lambda\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-36-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 37"><a href="#Final-Exam-Review-4-36" title="Copy heading and permalink for Exercise 37" aria-label="Copy heading and permalink for Exercise 37">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-37"><h4 class="heading"><span class="codenumber">38<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-37-1-1">Suppose a parameter <span class="process-math">\(\theta\)</span> takes values in <span class="process-math">\([0, 1]\)</span> with likelihood function <span class="process-math">\(\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2\text{.}\)</span> Find the maximum likelihood estimation of <span class="process-math">\(\theta\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-37-1-1" 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="Exercise 38"><a href="#Final-Exam-Review-4-37" title="Copy heading and permalink for Exercise 38" aria-label="Copy heading and permalink for Exercise 38">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-38"><h4 class="heading"><span class="codenumber">39<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-38-1-1">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="#Final-Exam-Review-4-38-1-1" 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="Final-Exam-Review-4-38-2" 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="Final-Exam-Review-4-38-2-1">
<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="#Final-Exam-Review-4-38-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 39.1"><a href="#Final-Exam-Review-4-38-2" title="Copy heading and permalink for Answer 39.1" aria-label="Copy heading and permalink for Answer 39.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 39"><a href="#Final-Exam-Review-4-38" title="Copy heading and permalink for Exercise 39" aria-label="Copy heading and permalink for Exercise 39">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-39"><h4 class="heading"><span class="codenumber">40<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-39-1-1">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="#Final-Exam-Review-4-39-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-39-2" 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="Final-Exam-Review-4-39-2-1">
<span class="process-math">\(\Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510.\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-39-2-1" 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 40.1"><a href="#Final-Exam-Review-4-39-2" title="Copy heading and permalink for Answer 40.1" aria-label="Copy heading and permalink for Answer 40.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 40"><a href="#Final-Exam-Review-4-39" title="Copy heading and permalink for Exercise 40" aria-label="Copy heading and permalink for Exercise 40">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-40"><h4 class="heading"><span class="codenumber">41<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-40-1"><div class="para" id="Final-Exam-Review-4-40-1-1">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="#Final-Exam-Review-4-40-1-1" 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="Final-Exam-Review-4-40-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-40-2-1-1">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="#Final-Exam-Review-4-40-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-40-2-2" 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="Final-Exam-Review-4-40-2-2-1">
<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="#Final-Exam-Review-4-40-2-2-1" 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 41.a.1"><a href="#Final-Exam-Review-4-40-2-2" title="Copy heading and permalink for Answer 41.a.1" aria-label="Copy heading and permalink for Answer 41.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 41.a"><a href="#Final-Exam-Review-4-40-2" title="Copy heading and permalink for Task 41.a" aria-label="Copy heading and permalink for Task 41.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-40-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-40-3-1-1">Estimate the probability that <span class="process-math">\(m \geq 69.5\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-40-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-40-3-2" 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="Final-Exam-Review-4-40-3-2-1">
<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="#Final-Exam-Review-4-40-3-2-1" 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 41.b.1"><a href="#Final-Exam-Review-4-40-3-2" title="Copy heading and permalink for Answer 41.b.1" aria-label="Copy heading and permalink for Answer 41.b.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 41.b"><a href="#Final-Exam-Review-4-40-3" title="Copy heading and permalink for Task 41.b" aria-label="Copy heading and permalink for Task 41.b">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-40-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-40-4-1-1">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="#Final-Exam-Review-4-40-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-40-4-2" 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="Final-Exam-Review-4-40-4-2-1">
<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="#Final-Exam-Review-4-40-4-2-1" 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 41.c.1"><a href="#Final-Exam-Review-4-40-4-2" title="Copy heading and permalink for Answer 41.c.1" aria-label="Copy heading and permalink for Answer 41.c.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 41.c"><a href="#Final-Exam-Review-4-40-4" title="Copy heading and permalink for Task 41.c" aria-label="Copy heading and permalink for Task 41.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 41"><a href="#Final-Exam-Review-4-40" title="Copy heading and permalink for Exercise 41" aria-label="Copy heading and permalink for Exercise 41">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-41"><h4 class="heading"><span class="codenumber">42<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-41-1-1">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="#Final-Exam-Review-4-41-1-1" 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="Final-Exam-Review-4-41-2" 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="Final-Exam-Review-4-41-2-1">
<span class="process-math">\([0.504, 0.696]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-41-2-1" 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 42.1"><a href="#Final-Exam-Review-4-41-2" title="Copy heading and permalink for Answer 42.1" aria-label="Copy heading and permalink for Answer 42.1">🔗</a></div>
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<div class="autopermalink" data-description="Exercise 42"><a href="#Final-Exam-Review-4-41" title="Copy heading and permalink for Exercise 42" aria-label="Copy heading and permalink for Exercise 42">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-42"><h4 class="heading"><span class="codenumber">43<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-42-1"><div class="para" id="Final-Exam-Review-4-42-1-1">Suppose in a sample of 100 people, 12 are left-handed.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-42-1-1" 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="Final-Exam-Review-4-42-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-42-2-1-1">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="#Final-Exam-Review-4-42-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="solutions"><details id="Final-Exam-Review-4-42-2-2" 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="Final-Exam-Review-4-42-2-2-1">
<span class="process-math">\([0.056, 0.184]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-42-2-2-1" 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 43.a.1"><a href="#Final-Exam-Review-4-42-2-2" title="Copy heading and permalink for Answer 43.a.1" aria-label="Copy heading and permalink for Answer 43.a.1">🔗</a></div>
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<div class="autopermalink" data-description="Task 43.a"><a href="#Final-Exam-Review-4-42-2" title="Copy heading and permalink for Task 43.a" aria-label="Copy heading and permalink for Task 43.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-42-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-42-3-1-1">Give a 90% confidence interval.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-42-3-1-1" 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="Final-Exam-Review-4-42-3-2" 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="Final-Exam-Review-4-42-3-2-1">
<span class="process-math">\([0.066, 0.174]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-42-3-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 43.b.1"><a href="#Final-Exam-Review-4-42-3-2" title="Copy heading and permalink for Answer 43.b.1" aria-label="Copy heading and permalink for Answer 43.b.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Task 43.b"><a href="#Final-Exam-Review-4-42-3" title="Copy heading and permalink for Task 43.b" aria-label="Copy heading and permalink for Task 43.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 43"><a href="#Final-Exam-Review-4-42" title="Copy heading and permalink for Exercise 43" aria-label="Copy heading and permalink for Exercise 43">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-43"><h4 class="heading"><span class="codenumber">44<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-43-1-1">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="#Final-Exam-Review-4-43-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-43-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">152<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 152: "><a href="#Final-Exam-Review-4-43-1-2" title="Copy heading and permalink for Table 152: " aria-label="Copy heading and permalink for Table 152: ">🔗</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="Final-Exam-Review-4-43-2" 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="Final-Exam-Review-4-43-2-1"> <span class="process-math">\([23.39, 32.61]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-43-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 44.1"><a href="#Final-Exam-Review-4-43-2" title="Copy heading and permalink for Answer 44.1" aria-label="Copy heading and permalink for Answer 44.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 44"><a href="#Final-Exam-Review-4-43" title="Copy heading and permalink for Exercise 44" aria-label="Copy heading and permalink for Exercise 44">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-44"><h4 class="heading"><span class="codenumber">45<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-44-1-1">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="#Final-Exam-Review-4-44-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-44-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">153<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 153: "><a href="#Final-Exam-Review-4-44-1-2" title="Copy heading and permalink for Table 153: " aria-label="Copy heading and permalink for Table 153: ">🔗</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="Final-Exam-Review-4-44-2" 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="Final-Exam-Review-4-44-2-1"> <span class="process-math">\([14.61, 19.39]\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-44-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Answer 45.1"><a href="#Final-Exam-Review-4-44-2" title="Copy heading and permalink for Answer 45.1" aria-label="Copy heading and permalink for Answer 45.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Exercise 45"><a href="#Final-Exam-Review-4-44" title="Copy heading and permalink for Exercise 45" aria-label="Copy heading and permalink for Exercise 45">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-45"><h4 class="heading"><span class="codenumber">46<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-45-1"><div class="para" id="Final-Exam-Review-4-45-1-1">In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-45-1-1" 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="Final-Exam-Review-4-45-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-45-2-1-1">We have a coin which we’ve flipped many times, seeing an above-average number of heads. We suspect the coin comes up heads more often than a fair coin would.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-45-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 46.a"><a href="#Final-Exam-Review-4-45-2" title="Copy heading and permalink for Task 46.a" aria-label="Copy heading and permalink for Task 46.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-45-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-45-3-1-1">We find a coin on the street and wonder whether or not it’s a fair coin.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-45-3-1-1" 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 46.b"><a href="#Final-Exam-Review-4-45-3" title="Copy heading and permalink for Task 46.b" aria-label="Copy heading and permalink for Task 46.b">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-45-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-45-4-1-1">We suspect there will be a difference in average weight of mice caught during the summer versus during the winter.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-45-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 46.c"><a href="#Final-Exam-Review-4-45-4" title="Copy heading and permalink for Task 46.c" aria-label="Copy heading and permalink for Task 46.c">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-45-5"><h5 class="heading"><span class="codenumber">(d)</span></h5>
<div class="para" id="Final-Exam-Review-4-45-5-1-1">In a medical study, a group of patients are gathered and the proportion experiencing particular symptoms is measured. A new drug intended to eliminate these symptoms is administered, after which the proportion experiencing symptoms is measured again.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-45-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 46.d"><a href="#Final-Exam-Review-4-45-5" title="Copy heading and permalink for Task 46.d" aria-label="Copy heading and permalink for Task 46.d">🔗</a></div></article><div class="autopermalink" data-description="Exercise 46"><a href="#Final-Exam-Review-4-45" title="Copy heading and permalink for Exercise 46" aria-label="Copy heading and permalink for Exercise 46">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-46"><h4 class="heading"><span class="codenumber">47<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-46-1-1">Suppose we find a coin and wonder whether it’s fair. As a first test, we decide to flip the coin 200 times and count the number of heads. If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-46-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 47"><a href="#Final-Exam-Review-4-46" title="Copy heading and permalink for Exercise 47" aria-label="Copy heading and permalink for Exercise 47">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-47"><h4 class="heading"><span class="codenumber">48<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-47-1-1">Suppose we have a coin which we suspect comes up heads more often than a fair coin would. As a first test, we decide to flip the coin 200 times and count the number of heads. If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-47-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 48"><a href="#Final-Exam-Review-4-47" title="Copy heading and permalink for Exercise 48" aria-label="Copy heading and permalink for Exercise 48">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-48"><h4 class="heading"><span class="codenumber">49<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-48-1-1">Suppose we find a six-sided die and wonder whether it’s fair. As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. If we roll 22 1’s, should we accept or reject the null hypothesis of a fair die at a significance level of 0.05?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-48-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 49"><a href="#Final-Exam-Review-4-48" title="Copy heading and permalink for Exercise 49" aria-label="Copy heading and permalink for Exercise 49">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-49"><h4 class="heading"><span class="codenumber">50<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-49-1-1">Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<span class="process-math">\(^2\text{.}\)</span> We suspect that growing this plant in a greenhouse will increase its height. A sample of 50 plants grown in a greenhouse has an average height of 40 in. Is this significant enough data to reject the null hypothesis of equal means at the <span class="process-math">\(p = 0.05\)</span> significance level?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-49-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 50"><a href="#Final-Exam-Review-4-49" title="Copy heading and permalink for Exercise 50" aria-label="Copy heading and permalink for Exercise 50">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-50"><h4 class="heading"><span class="codenumber">51<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-50-1-1">A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety. A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams<span class="process-math">\(^2\)</span> (for an individual apple). An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams<span class="process-math">\(^2\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-50-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 51"><a href="#Final-Exam-Review-4-50" title="Copy heading and permalink for Exercise 51" aria-label="Copy heading and permalink for Exercise 51">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-51"><h4 class="heading"><span class="codenumber">52<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-51-1-1">We have an established factory which produces coins that are close to fair. We’re opening up a second factory, and we’d like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. We pick one sample coin from each factory, and flip each sample coin 100 times. The coin from the established factory flips 52 heads in 100 flips. The coin from the new factory flips 62 heads in 100 flips. Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-51-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 52"><a href="#Final-Exam-Review-4-51" title="Copy heading and permalink for Exercise 52" aria-label="Copy heading and permalink for Exercise 52">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-52"><h4 class="heading"><span class="codenumber">53<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-52-1-1">Suppose we find a coin and wonder whether it’s fair. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-52-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 53"><a href="#Final-Exam-Review-4-52" title="Copy heading and permalink for Exercise 53" aria-label="Copy heading and permalink for Exercise 53">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-53"><h4 class="heading"><span class="codenumber">54<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-53-1-1">Suppose we have a coin which we suspect comes up heads more often than a fair coin would. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-53-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 54"><a href="#Final-Exam-Review-4-53" title="Copy heading and permalink for Exercise 54" aria-label="Copy heading and permalink for Exercise 54">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-54"><h4 class="heading"><span class="codenumber">55<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-54-1"><div class="para" id="Final-Exam-Review-4-54-1-1">Suppose we find a six-sided die and wonder whether it’s fair. As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. The expected number of 1’s is 50/3, with a variance of 125/9.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-54-1-1" 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="Final-Exam-Review-4-54-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-54-2-1-1">Using a normal approximation, what is the smallest number of 1’s greater than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-54-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 55.a"><a href="#Final-Exam-Review-4-54-2" title="Copy heading and permalink for Task 55.a" aria-label="Copy heading and permalink for Task 55.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-54-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-54-3-1-1">Using a normal approximation, what is the greatest number of 1’s less than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-54-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 55.b"><a href="#Final-Exam-Review-4-54-3" title="Copy heading and permalink for Task 55.b" aria-label="Copy heading and permalink for Task 55.b">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-54-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-54-4-1-1">Suppose that this die is weighted so that it rolls a 1 with probability 0.2. What would be the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-54-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 55.c"><a href="#Final-Exam-Review-4-54-4" title="Copy heading and permalink for Task 55.c" aria-label="Copy heading and permalink for Task 55.c">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-54-5"><h5 class="heading"><span class="codenumber">(d)</span></h5>
<div class="para" id="Final-Exam-Review-4-54-5-1-1">Suppose we roll the die 100 times and see 23 1’s. Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-54-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 55.d"><a href="#Final-Exam-Review-4-54-5" title="Copy heading and permalink for Task 55.d" aria-label="Copy heading and permalink for Task 55.d">🔗</a></div></article><div class="autopermalink" data-description="Exercise 55"><a href="#Final-Exam-Review-4-54" title="Copy heading and permalink for Exercise 55" aria-label="Copy heading and permalink for Exercise 55">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-55"><h4 class="heading"><span class="codenumber">56<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-55-1-1">Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<span class="process-math">\(^2\text{.}\)</span> We suspect that growing this plant in a greenhouse will increase its height. We take the average height of a sample of 50 plants grown in a greenhouse. What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the <span class="process-math">\(p = 0.05\)</span> significance level? If the plants, when grown in a greenhouse, would truly have an average height of 41 in, what is the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-55-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 56"><a href="#Final-Exam-Review-4-55" title="Copy heading and permalink for Exercise 56" aria-label="Copy heading and permalink for Exercise 56">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-56"><h4 class="heading"><span class="codenumber">57<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-56-1-1">A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. They survey their customers about their favorite drinks. Is the data below consistent with the null hypothesis that each type of drink will be equally preferred?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-56-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-56-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">154<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 154: "><a href="#Final-Exam-Review-4-56-1-2" title="Copy heading and permalink for Table 154: " aria-label="Copy heading and permalink for Table 154: ">🔗</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">drink type</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">water</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">soda</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">tea</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">coffee</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">energy drinks</th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">favorite</td>
<td class="c m b0 r0 l0 t0 lines">28</td>
<td class="c m b0 r0 l0 t0 lines">17</td>
<td class="c m b0 r0 l0 t0 lines">15</td>
<td class="c m b0 r0 l0 t0 lines">26</td>
<td class="c m b0 r0 l0 t0 lines">14</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 57"><a href="#Final-Exam-Review-4-56" title="Copy heading and permalink for Exercise 57" aria-label="Copy heading and permalink for Exercise 57">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-57"><h4 class="heading"><span class="codenumber">58<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="introduction" id="Final-Exam-Review-4-57-1">
<div class="para" id="Final-Exam-Review-4-57-1-1">A particular drug is administered in 100 independent trials. In each trial, the drug is administered to four people, and we count how many respond to the drug. The table below shows how many trials have each different count of people who respond to the drug.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-57-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<figure class="table table-like" id="Final-Exam-Review-4-57-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">155<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 155: "><a href="#Final-Exam-Review-4-57-1-2" title="Copy heading and permalink for Table 155: " aria-label="Copy heading and permalink for Table 155: ">🔗</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"># who respond to drug</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">0</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">1</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">2</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">3</th>
<th scope="col" class="c m b1 r0 l0 t0 lines">4</th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines"># of trials</td>
<td class="c m b0 r0 l0 t0 lines">3</td>
<td class="c m b0 r0 l0 t0 lines">11</td>
<td class="c m b0 r0 l0 t0 lines">31</td>
<td class="c m b0 r0 l0 t0 lines">34</td>
<td class="c m b0 r0 l0 t0 lines">21</td>
</tr>
</table></div></figure>
</div>
<article class="task exercise-like" id="Final-Exam-Review-4-57-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
<div class="para" id="Final-Exam-Review-4-57-2-1-1">Is the data consistent with a binomial distribution with parameter <span class="process-math">\(\theta = 0.7\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-57-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 58.a"><a href="#Final-Exam-Review-4-57-2" title="Copy heading and permalink for Task 58.a" aria-label="Copy heading and permalink for Task 58.a">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-57-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
<div class="para" id="Final-Exam-Review-4-57-3-1-1">What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation <span class="process-math">\(\widehat{\theta}\)</span> for the probability that a person will respond to the drug?<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-57-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 58.b"><a href="#Final-Exam-Review-4-57-3" title="Copy heading and permalink for Task 58.b" aria-label="Copy heading and permalink for Task 58.b">🔗</a></div></article><article class="task exercise-like" id="Final-Exam-Review-4-57-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
<div class="para" id="Final-Exam-Review-4-57-4-1-1">Is the data consistent with a binomial distribution with the MLE value of <span class="process-math">\(\widehat{\theta}\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-57-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Task 58.c"><a href="#Final-Exam-Review-4-57-4" title="Copy heading and permalink for Task 58.c" aria-label="Copy heading and permalink for Task 58.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 58"><a href="#Final-Exam-Review-4-57" title="Copy heading and permalink for Exercise 58" aria-label="Copy heading and permalink for Exercise 58">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-58"><h4 class="heading"><span class="codenumber">59<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-58-1-1">Calculate the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-58-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-58-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">156<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 156: "><a href="#Final-Exam-Review-4-58-1-2" title="Copy heading and permalink for Table 156: " aria-label="Copy heading and permalink for Table 156: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.12</td>
<td class="c m b1 r0 l0 t0 lines">0.24</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.3</td>
<td class="c m b0 r0 l0 t0 lines">34</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 59"><a href="#Final-Exam-Review-4-58" title="Copy heading and permalink for Exercise 59" aria-label="Copy heading and permalink for Exercise 59">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-59"><h4 class="heading"><span class="codenumber">60<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-59-1-1">Calculate the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-59-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-59-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">157<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 157: "><a href="#Final-Exam-Review-4-59-1-2" title="Copy heading and permalink for Table 157: " aria-label="Copy heading and permalink for Table 157: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.08</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.14</td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.32</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 60"><a href="#Final-Exam-Review-4-59" title="Copy heading and permalink for Exercise 60" aria-label="Copy heading and permalink for Exercise 60">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-60"><h4 class="heading"><span class="codenumber">61<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-60-1-1">Suppose we roll a fair, 4-sided die two times. Let <span class="process-math">\(X\)</span> be the sum of the rolls, and let <span class="process-math">\(Y\)</span> be the product of the rolls. Find the covariance of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-60-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="Final-Exam-Review-4-60-1-2">[Note: Somewhat tedious.]<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-60-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 61"><a href="#Final-Exam-Review-4-60" title="Copy heading and permalink for Exercise 61" aria-label="Copy heading and permalink for Exercise 61">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-61"><h4 class="heading"><span class="codenumber">62<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-61-1-1">Calculate the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-61-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-61-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">158<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 158: "><a href="#Final-Exam-Review-4-61-1-2" title="Copy heading and permalink for Table 158: " aria-label="Copy heading and permalink for Table 158: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.12</td>
<td class="c m b1 r0 l0 t0 lines">0.24</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.3</td>
<td class="c m b0 r0 l0 t0 lines">34</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 62"><a href="#Final-Exam-Review-4-61" title="Copy heading and permalink for Exercise 62" aria-label="Copy heading and permalink for Exercise 62">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-62"><h4 class="heading"><span class="codenumber">63<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-62-1-1">Calculate the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-62-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-62-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">159<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 159: "><a href="#Final-Exam-Review-4-62-1-2" title="Copy heading and permalink for Table 159: " aria-label="Copy heading and permalink for Table 159: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</tr>
<tr>
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
<td class="c m b1 r1 l0 t0 lines">0.08</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.1</td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
<td class="c m b0 r1 l0 t0 lines">0.14</td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.32</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 63"><a href="#Final-Exam-Review-4-62" title="Copy heading and permalink for Exercise 63" aria-label="Copy heading and permalink for Exercise 63">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-63"><h4 class="heading"><span class="codenumber">64<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-63-1-1">Suppose we roll a fair, 4-sided die two times. Let <span class="process-math">\(X\)</span> be the sum of the rolls, and let <span class="process-math">\(Y\)</span> be the product of the rolls. Find the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-63-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="Final-Exam-Review-4-63-1-2">[Note: Somewhat tedious.]<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-63-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 64"><a href="#Final-Exam-Review-4-63" title="Copy heading and permalink for Exercise 64" aria-label="Copy heading and permalink for Exercise 64">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-64"><h4 class="heading"><span class="codenumber">65<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-64-1-1">A survey of local companies collects information about marketing budgets <span class="process-math">\(X\)</span> and revenue <span class="process-math">\(Y\)</span> (each measures in thousands of dollars), shown below. A linear regression gives the best linear fit as <span class="process-math">\(Y = 18.28 X + 29.69\text{.}\)</span> What is the coefficient of determination <span class="process-math">\(r^2\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-64-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-64-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">160<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 160: "><a href="#Final-Exam-Review-4-64-1-2" title="Copy heading and permalink for Table 160: " aria-label="Copy heading and permalink for Table 160: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="r m b1 r0 l0 t0 lines"><span class="process-math">\(x_i\)</span></th>
<th scope="col" class="r m b1 r0 l0 t0 lines"><span class="process-math">\(y_i\)</span></th>
<th scope="col" class="r m b1 r0 l0 t0 lines"><span class="process-math">\((y_i - \text{avg})^2\)</span></th>
<th scope="col" class="r m b1 r0 l0 t0 lines"><span class="process-math">\(\text{pred } y_i\)</span></th>
<th scope="col" class="r m b1 r0 l0 t0 lines"><span class="process-math">\(\text{res}^2\)</span></th>
</tr>
<tr>
<td class="r m b0 r0 l0 t0 lines">200</td>
<td class="r m b0 r0 l0 t0 lines">4300</td>
<td class="r m b0 r0 l0 t0 lines">2073600</td>
<td class="r m b0 r0 l0 t0 lines">3686</td>
<td class="r m b0 r0 l0 t0 lines">377377</td>
</tr>
<tr>
<td class="r m b0 r0 l0 t0 lines">420</td>
<td class="r m b0 r0 l0 t0 lines">7700</td>
<td class="r m b0 r0 l0 t0 lines">3841600</td>
<td class="r m b0 r0 l0 t0 lines">7707</td>
<td class="r m b0 r0 l0 t0 lines">53</td>
</tr>
<tr>
<td class="r m b0 r0 l0 t0 lines">270</td>
<td class="r m b0 r0 l0 t0 lines">4500</td>
<td class="r m b0 r0 l0 t0 lines">1537600</td>
<td class="r m b0 r0 l0 t0 lines">4965</td>
<td class="r m b0 r0 l0 t0 lines">216495</td>
</tr>
<tr>
<td class="r m b0 r0 l0 t0 lines">380</td>
<td class="r m b0 r0 l0 t0 lines">7000</td>
<td class="r m b0 r0 l0 t0 lines">1587600</td>
<td class="r m b0 r0 l0 t0 lines">6976</td>
<td class="r m b0 r0 l0 t0 lines">572</td>
</tr>
<tr>
<td class="r m b1 r0 l0 t0 lines">300</td>
<td class="r m b1 r0 l0 t0 lines">5200</td>
<td class="r m b1 r0 l0 t0 lines">291600</td>
<td class="r m b1 r0 l0 t0 lines">5514</td>
<td class="r m b1 r0 l0 t0 lines">98401</td>
</tr>
<tr>
<td class="r m b0 r0 l0 t0 lines">sum:</td>
<td class="r m b0 r0 l0 t0 lines">28700</td>
<td class="r m b0 r0 l0 t0 lines">9332000</td>
<td class="r m b0 r0 l0 t0 lines">28848</td>
<td class="r m b0 r0 l0 t0 lines">692898</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 65"><a href="#Final-Exam-Review-4-64" title="Copy heading and permalink for Exercise 65" aria-label="Copy heading and permalink for Exercise 65">🔗</a></div></article><article class="exercise exercise-like" id="Final-Exam-Review-4-65"><h4 class="heading"><span class="codenumber">66<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Final-Exam-Review-4-65-1-1">A sample of 100 measurements are taken and a best fit line is calculated, resulting in the data below (the final line shows the sums for each column). Find the coefficient of determination.<div class="autopermalink" data-description="Paragraph"><a href="#Final-Exam-Review-4-65-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Final-Exam-Review-4-65-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">161<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 161: "><a href="#Final-Exam-Review-4-65-1-2" title="Copy heading and permalink for Table 161: " aria-label="Copy heading and permalink for Table 161: ">🔗</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"><span class="process-math">\(x_i\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(y_i\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\text{pred } y_i\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\((y - \text{avg})^2\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\text{res}^2\)</span></th>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">38.00</td>
<td class="c m b0 r0 l0 t0 lines">121.00</td>
<td class="c m b0 r0 l0 t0 lines">93.17</td>
<td class="c m b0 r0 l0 t0 lines">11.83</td>
<td class="c m b0 r0 l0 t0 lines">774.33</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">87.00</td>
<td class="c m b0 r0 l0 t0 lines">241.00</td>
<td class="c m b0 r0 l0 t0 lines">238.95</td>
<td class="c m b0 r0 l0 t0 lines">13586.23</td>
<td class="c m b0 r0 l0 t0 lines">4.22</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">30.00</td>
<td class="c m b0 r0 l0 t0 lines">61.00</td>
<td class="c m b0 r0 l0 t0 lines">69.37</td>
<td class="c m b0 r0 l0 t0 lines">4024.63</td>
<td class="c m b0 r0 l0 t0 lines">70.12</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">35.00</td>
<td class="c m b0 r0 l0 t0 lines">85.00</td>
<td class="c m b0 r0 l0 t0 lines">84.25</td>
<td class="c m b0 r0 l0 t0 lines">1555.51</td>
<td class="c m b0 r0 l0 t0 lines">0.57</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">⋮</td>
<td class="c m b0 r0 l0 t0 lines">⋮</td>
<td class="c m b0 r0 l0 t0 lines">⋮</td>
<td class="c m b0 r0 l0 t0 lines">⋮</td>
<td class="c m b0 r0 l0 t0 lines">⋮</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">33.00</td>
<td class="c m b0 r0 l0 t0 lines">108.00</td>
<td class="c m b0 r0 l0 t0 lines">78.30</td>
<td class="c m b0 r0 l0 t0 lines">270.27</td>
<td class="c m b0 r0 l0 t0 lines">882.19</td>
</tr>
<tr>
<td class="c m b1 r0 l0 t0 lines">26.00</td>
<td class="c m b1 r0 l0 t0 lines">32.00</td>
<td class="c m b1 r0 l0 t0 lines">57.47</td>
<td class="c m b1 r0 l0 t0 lines">8545.15</td>
<td class="c m b1 r0 l0 t0 lines">648.91</td>
</tr>
<tr>
<td class="c m b0 r0 l0 t0 lines">sum:</td>
<td class="c m b0 r0 l0 t0 lines">12444.00</td>
<td class="c m b0 r0 l0 t0 lines">12444.00</td>
<td class="c m b0 r0 l0 t0 lines">515206.64</td>
<td class="c m b0 r0 l0 t0 lines">29789.80</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 66"><a href="#Final-Exam-Review-4-65" title="Copy heading and permalink for Exercise 66" aria-label="Copy heading and permalink for Exercise 66">🔗</a></div></article><div class="autopermalink" data-description="Exercises"><a href="#Final-Exam-Review-4" title="Copy heading and permalink for Exercises" aria-label="Copy heading and permalink for Exercises">🔗</a></div></section><div class="autopermalink" data-description="Section: Final Exam Review"><a href="#Final-Exam-Review" title="Copy heading and permalink for Section: Final Exam Review" aria-label="Copy heading and permalink for Section: Final Exam Review">🔗</a></div></section></div>
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