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course/.mapping.json
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{"source/main.ptx": ["frontmatter", "course-documents", "ch-notes", "quizzes", "exams", "recitations", "review", "fun"], "source/syllabus/syllabus.ptx": ["syllabus"], "source/notes/1-13.ptx": ["notes-01-13"], "source/notes/1-15.ptx": ["notes-01-15"], "source/notes/1-20.ptx": ["notes-01-20"], "source/notes/1-22.ptx": ["notes-01-22"], "source/notes/1-27.ptx": ["notes-01-27"], "source/notes/1-29.ptx": ["notes-01-29"], "source/notes/2-3.ptx": ["notes-02-03"], "source/notes/2-5.ptx": ["notes-02-05"], "source/notes/2-10.ptx": ["notes-02-10"], "source/notes/2-19.ptx": ["notes-02-19"], "source/notes/2-24.ptx": ["notes-02-24"], "source/quizzes/quiz-01.ptx": ["quiz-01"], "source/quizzes/quiz-02.ptx": ["quiz-02"], "source/quizzes/quiz-03.ptx": ["quiz-03"], "source/quizzes/quiz-04.ptx": ["quiz-04"], "source/exams/exam-01.ptx": ["exam-01"], "source/exams/exam-02.ptx": ["exam-02"], "source/recitations/calculus-review.ptx": ["recitation-calculus-review"], "source/review/Exam-1-Review.ptx": ["Exam-1-Review"], "source/review/Exam-2-Review.ptx": ["Exam-2-Review"], "source/review/Final-Exam-Review.ptx": ["Final-Exam-Review"], "source/fun/boxes.ptx": ["boxes"]} {"source/main.ptx": ["frontmatter", "course-documents", "ch-notes", "quizzes", "exams", "recitations", "review", "fun"], "source/syllabus/syllabus.ptx": ["syllabus"], "source/notes/1-13.ptx": ["notes-01-13"], "source/notes/1-15.ptx": ["notes-01-15"], "source/notes/1-20.ptx": ["notes-01-20"], "source/notes/1-22.ptx": ["notes-01-22"], "source/notes/1-27.ptx": ["notes-01-27"], "source/notes/1-29.ptx": ["notes-01-29"], "source/notes/2-3.ptx": ["notes-02-03"], "source/notes/2-5.ptx": ["notes-02-05"], "source/notes/2-10.ptx": ["notes-02-10"], "source/notes/2-19.ptx": ["notes-02-19"], "source/notes/2-24.ptx": ["notes-02-24"], "source/quizzes/quiz-01.ptx": ["quiz-01"], "source/quizzes/quiz-02.ptx": ["quiz-02"], "source/quizzes/quiz-03.ptx": ["quiz-03"], "source/quizzes/quiz-04.ptx": ["quiz-04"], "source/quizzes/quiz-05.ptx": ["quiz-05"], "source/exams/exam-01.ptx": ["exam-01"], "source/exams/exam-02.ptx": ["exam-02"], "source/recitations/calculus-review.ptx": ["recitation-calculus-review"], "source/review/Exam-1-Review.ptx": ["Exam-1-Review"], "source/review/Exam-2-Review.ptx": ["Exam-2-Review"], "source/review/Final-Exam-Review.ptx": ["Final-Exam-Review"], "source/fun/boxes.ptx": ["boxes"]}
course/Exam-1-Review.html
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@@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
@@ -394,7 +395,7 @@ eBookConfig.enable_chatcodes = false;
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Exercise 4"><a href="#Exam-1-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="Exam-1-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="autopermalink" data-description="Exercise 4"><a href="#Exam-1-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="Exam-1-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="Exam-1-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="#Exam-1-Review-4-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-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="Exam-1-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">128<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 128: "><a href="#Exam-1-Review-4-5-1-2" title="Copy heading and permalink for Table 128: " aria-label="Copy heading and permalink for Table 128: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-1-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">132<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 132: "><a href="#Exam-1-Review-4-5-1-2" title="Copy heading and permalink for Table 132: " aria-label="Copy heading and permalink for Table 132: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -445,7 +446,7 @@ eBookConfig.enable_chatcodes = false;
<div class="para" id="Exam-1-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="#Exam-1-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="solutions"><details id="Exam-1-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"> <div class="solutions"><details id="Exam-1-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="Exam-1-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">129<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 129: Probability Distribution for a Linearly Scaled Die"><a href="#Exam-1-Review-4-6-2-1" title="Copy heading and permalink for Table 129: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 129: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <figure class="table table-like" id="Exam-1-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">133<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 133: Probability Distribution for a Linearly Scaled Die"><a href="#Exam-1-Review-4-6-2-1" title="Copy heading and permalink for Table 133: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 133: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -480,7 +481,7 @@ eBookConfig.enable_chatcodes = false;
<div class="para" id="Exam-1-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="#Exam-1-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="solutions"><details id="Exam-1-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"> <div class="solutions"><details id="Exam-1-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="Exam-1-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">130<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 130: Probability Distribution for an Even-biased Die"><a href="#Exam-1-Review-4-7-2-1" title="Copy heading and permalink for Table 130: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 130: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <figure class="table table-like" id="Exam-1-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">134<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 134: Probability Distribution for an Even-biased Die"><a href="#Exam-1-Review-4-7-2-1" title="Copy heading and permalink for Table 134: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 134: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -674,7 +675,7 @@ A\cap B \amp = \{HHH, HHT\}
<div class="para" id="Exam-1-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="#Exam-1-Review-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-Review-4-13-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="solutions"><details id="Exam-1-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"> <div class="solutions"><details id="Exam-1-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="Exam-1-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">131<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 131: Example Distribution"><a href="#Exam-1-Review-4-13-2-1" title="Copy heading and permalink for Table 131: Example Distribution" aria-label="Copy heading and permalink for Table 131: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <figure class="table table-like" id="Exam-1-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">135<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 135: Example Distribution"><a href="#Exam-1-Review-4-13-2-1" title="Copy heading and permalink for Table 135: Example Distribution" aria-label="Copy heading and permalink for Table 135: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -799,7 +800,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Exercise 20"><a href="#Exam-1-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="Exam-1-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="autopermalink" data-description="Exercise 20"><a href="#Exam-1-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="Exam-1-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="Exam-1-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="#Exam-1-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-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="Exam-1-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">132<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 132: Joint distribution for X, Y"><a href="#Exam-1-Review-4-20-1-2" title="Copy heading and permalink for Table 132: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 132: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-1-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">136<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 136: Joint distribution for X, Y"><a href="#Exam-1-Review-4-20-1-2" title="Copy heading and permalink for Table 136: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 136: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <tr>
<td class="c m b1 r1 l0 t0 lines"></td> <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">\(X = 0\)</span></td>
@@ -834,7 +835,7 @@ A\cap B \amp = \{HHH, HHT\}
<div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-21-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="autopermalink" data-description="Paragraph"><a href="#Exam-1-Review-4-21-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<div class="solutions"><details id="Exam-1-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"> <div class="solutions"><details id="Exam-1-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="Exam-1-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">133<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 133: Joint Distribution"><a href="#Exam-1-Review-4-21-2-1" title="Copy heading and permalink for Table 133: Joint Distribution" aria-label="Copy heading and permalink for Table 133: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <figure class="table table-like" id="Exam-1-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">137<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 137: Joint Distribution"><a href="#Exam-1-Review-4-21-2-1" title="Copy heading and permalink for Table 137: Joint Distribution" aria-label="Copy heading and permalink for Table 137: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr> <tr>
<td class="c m b1 r1 l0 t0 lines"></td> <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">\(X = 0\)</span></td>
@@ -857,7 +858,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Exercise 22"><a href="#Exam-1-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="Exam-1-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="autopermalink" data-description="Exercise 22"><a href="#Exam-1-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="Exam-1-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="Exam-1-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="#Exam-1-Review-4-22-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-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="Exam-1-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">134<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 134: "><a href="#Exam-1-Review-4-22-1-2" title="Copy heading and permalink for Table 134: " aria-label="Copy heading and permalink for Table 134: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-1-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">138<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 138: "><a href="#Exam-1-Review-4-22-1-2" title="Copy heading and permalink for Table 138: " aria-label="Copy heading and permalink for Table 138: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <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">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th> <th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
@@ -966,7 +967,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Exercise 28"><a href="#Exam-1-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="Exam-1-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="autopermalink" data-description="Exercise 28"><a href="#Exam-1-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="Exam-1-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="Exam-1-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="#Exam-1-Review-4-28-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-1-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="#Exam-1-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="Exam-1-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">135<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 135: "><a href="#Exam-1-Review-4-28-1-2" title="Copy heading and permalink for Table 135: " aria-label="Copy heading and permalink for Table 135: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-1-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">139<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 139: "><a href="#Exam-1-Review-4-28-1-2" title="Copy heading and permalink for Table 139: " aria-label="Copy heading and permalink for Table 139: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <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">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th> <th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
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<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
@@ -390,7 +391,7 @@ eBookConfig.enable_chatcodes = false;
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Task 5.b"><a href="#Exam-2-Review-4-5-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 5"><a href="#Exam-2-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="Exam-2-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="autopermalink" data-description="Task 5.b"><a href="#Exam-2-Review-4-5-3" title="Copy heading and permalink for Task 5.b" aria-label="Copy heading and permalink for Task 5.b">🔗</a></div></article><div class="autopermalink" data-description="Exercise 5"><a href="#Exam-2-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="Exam-2-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="Exam-2-Review-4-6-1-1">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="#Exam-2-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-2-Review-4-6-1-1">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="#Exam-2-Review-4-6-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-2-Review-4-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">136<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 136: "><a href="#Exam-2-Review-4-6-1-2" title="Copy heading and permalink for Table 136: " aria-label="Copy heading and permalink for Table 136: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-2-Review-4-6-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">140<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 140: "><a href="#Exam-2-Review-4-6-1-2" title="Copy heading and permalink for Table 140: " aria-label="Copy heading and permalink for Table 140: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines">mouse <span class="process-math">\(i\)</span> <th scope="col" class="c m b1 r0 l0 t0 lines">mouse <span class="process-math">\(i\)</span>
</th> </th>
@@ -423,7 +424,7 @@ eBookConfig.enable_chatcodes = false;
</div></details></div> </div></details></div>
<div class="autopermalink" data-description="Exercise 6"><a href="#Exam-2-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="Exam-2-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="autopermalink" data-description="Exercise 6"><a href="#Exam-2-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="Exam-2-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="Exam-2-Review-4-7-1-1">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="#Exam-2-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-2-Review-4-7-1-1">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="#Exam-2-Review-4-7-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-2-Review-4-7-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">137<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 137: "><a href="#Exam-2-Review-4-7-1-2" title="Copy heading and permalink for Table 137: " aria-label="Copy heading and permalink for Table 137: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-2-Review-4-7-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">141<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 141: "><a href="#Exam-2-Review-4-7-1-2" title="Copy heading and permalink for Table 141: " aria-label="Copy heading and permalink for Table 141: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines">plant <span class="process-math">\(i\)</span> <th scope="col" class="c m b1 r0 l0 t0 lines">plant <span class="process-math">\(i\)</span>
</th> </th>
@@ -513,7 +514,7 @@ eBookConfig.enable_chatcodes = false;
</div> </div>
<div class="autopermalink" data-description="Exercise 18"><a href="#Exam-2-Review-4-18" 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="Exam-2-Review-4-19"><h4 class="heading"><span class="codenumber">19<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4> <div class="autopermalink" data-description="Exercise 18"><a href="#Exam-2-Review-4-18" 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="Exam-2-Review-4-19"><h4 class="heading"><span class="codenumber">19<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="Exam-2-Review-4-19-1-1">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="#Exam-2-Review-4-19-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-2-Review-4-19-1-1">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="#Exam-2-Review-4-19-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="Exam-2-Review-4-19-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">138<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 138: "><a href="#Exam-2-Review-4-19-1-2" title="Copy heading and permalink for Table 138: " aria-label="Copy heading and permalink for Table 138: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </div> <figure class="table table-like" id="Exam-2-Review-4-19-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">142<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 142: "><a href="#Exam-2-Review-4-19-1-2" title="Copy heading and permalink for Table 142: " aria-label="Copy heading and permalink for Table 142: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <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">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">water</th>
@@ -534,7 +535,7 @@ eBookConfig.enable_chatcodes = false;
<div class="introduction" id="Exam-2-Review-4-20-1"> <div class="introduction" id="Exam-2-Review-4-20-1">
<div class="para" id="Exam-2-Review-4-20-1-1">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="#Exam-2-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div> <div class="para" id="Exam-2-Review-4-20-1-1">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="#Exam-2-Review-4-20-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> </div>
<figure class="table table-like" id="Exam-2-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">139<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 139: "><a href="#Exam-2-Review-4-20-1-2" title="Copy heading and permalink for Table 139: " aria-label="Copy heading and permalink for Table 139: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <figure class="table table-like" id="Exam-2-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">143<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 143: "><a href="#Exam-2-Review-4-20-1-2" title="Copy heading and permalink for Table 143: " aria-label="Copy heading and permalink for Table 143: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal"> <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"># 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">0</th>
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<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
@@ -394,7 +395,7 @@ eBookConfig.enable_chatcodes = false;
</div></details></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="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 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">140<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 140: "><a href="#Final-Exam-Review-4-5-1-2" title="Copy heading and permalink for Table 140: " aria-label="Copy heading and permalink for Table 140: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -445,7 +446,7 @@ eBookConfig.enable_chatcodes = false;
<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 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>
<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"> <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">141<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 141: Probability Distribution for a Linearly Scaled Die"><a href="#Final-Exam-Review-4-6-2-1" title="Copy heading and permalink for Table 141: Probability Distribution for a Linearly Scaled Die" aria-label="Copy heading and permalink for Table 141: Probability Distribution for a Linearly Scaled Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <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> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -480,7 +481,7 @@ eBookConfig.enable_chatcodes = false;
<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 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>
<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"> <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">142<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 142: Probability Distribution for an Even-biased Die"><a href="#Final-Exam-Review-4-7-2-1" title="Copy heading and permalink for Table 142: Probability Distribution for an Even-biased Die" aria-label="Copy heading and permalink for Table 142: Probability Distribution for an Even-biased Die">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <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> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -674,7 +675,7 @@ A\cap B \amp = \{HHH, HHT\}
<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 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>
<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"> <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">143<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 143: Example Distribution"><a href="#Final-Exam-Review-4-13-2-1" title="Copy heading and permalink for Table 143: Example Distribution" aria-label="Copy heading and permalink for Table 143: Example Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <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> <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">\(x\)</span></td>
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td> <td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(x)\)</span></td>
@@ -799,7 +800,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></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="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 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">144<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 144: Joint distribution for X, Y"><a href="#Final-Exam-Review-4-20-1-2" title="Copy heading and permalink for Table 144: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 144: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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> <tr>
<td class="c m b1 r1 l0 t0 lines"></td> <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">\(X = 0\)</span></td>
@@ -834,7 +835,7 @@ A\cap B \amp = \{HHH, HHT\}
<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 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>
<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"> <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">145<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 145: Joint Distribution"><a href="#Final-Exam-Review-4-21-2-1" title="Copy heading and permalink for Table 145: Joint Distribution" aria-label="Copy heading and permalink for Table 145: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> <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> <tr>
<td class="c m b1 r1 l0 t0 lines"></td> <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">\(X = 0\)</span></td>
@@ -857,7 +858,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></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="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 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">146<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 146: "><a href="#Final-Exam-Review-4-22-1-2" title="Copy heading and permalink for Table 146: " aria-label="Copy heading and permalink for Table 146: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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"> <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">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th> <th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
@@ -966,7 +967,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></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="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 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">147<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 147: "><a href="#Final-Exam-Review-4-28-1-2" title="Copy heading and permalink for Table 147: " aria-label="Copy heading and permalink for Table 147: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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"> <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">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th> <th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></th>
@@ -1114,7 +1115,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></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="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 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">148<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 148: "><a href="#Final-Exam-Review-4-43-1-2" title="Copy heading and permalink for Table 148: " aria-label="Copy heading and permalink for Table 148: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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"> <tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines">mouse <span class="process-math">\(i\)</span> <th scope="col" class="c m b1 r0 l0 t0 lines">mouse <span class="process-math">\(i\)</span>
</th> </th>
@@ -1147,7 +1148,7 @@ A\cap B \amp = \{HHH, HHT\}
</div></details></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="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 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">149<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 149: "><a href="#Final-Exam-Review-4-44-1-2" title="Copy heading and permalink for Table 149: " aria-label="Copy heading and permalink for Table 149: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular"> </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"> <tr class="header-horizontal">
<th scope="col" class="c m b1 r0 l0 t0 lines">plant <span class="process-math">\(i\)</span> <th scope="col" class="c m b1 r0 l0 t0 lines">plant <span class="process-math">\(i\)</span>
</th> </th>
@@ -1237,7 +1238,7 @@ A\cap B \amp = \{HHH, HHT\}
</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="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 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">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-4-56-1-2" 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"> </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"> <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">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">water</th>
@@ -1258,7 +1259,7 @@ A\cap B \amp = \{HHH, HHT\}
<div class="introduction" id="Final-Exam-Review-4-57-1"> <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 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> </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">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-4-57-1-2" 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"> <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"> <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"># 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">0</th>
@@ -1288,7 +1289,7 @@ A\cap B \amp = \{HHH, HHT\}
</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="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 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">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-4-58-1-2" 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"> </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"> <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"></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 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
@@ -1306,7 +1307,7 @@ A\cap B \amp = \{HHH, HHT\}
</tr> </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> </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 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">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-4-59-1-2" 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"> </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"> <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"></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">\(X = 0\)</span></th>
@@ -1331,7 +1332,7 @@ A\cap B \amp = \{HHH, HHT\}
</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="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 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">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-4-61-1-2" 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"> </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"> <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"></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 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
@@ -1349,7 +1350,7 @@ A\cap B \amp = \{HHH, HHT\}
</tr> </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> </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 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">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-4-62-1-2" 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"> </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"> <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"></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">\(X = 0\)</span></th>
@@ -1374,7 +1375,7 @@ A\cap B \amp = \{HHH, HHT\}
</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="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 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">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-4-64-1-2" 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"> </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"> <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">\(x_i\)</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">\(y_i\)</span></th>
@@ -1426,7 +1427,7 @@ A\cap B \amp = \{HHH, HHT\}
</tr> </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> </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 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">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-4-65-1-2" 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"> </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"> <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">\(x_i\)</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">\(y_i\)</span></th>
course/boxes.html
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<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
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<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
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@@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
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<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
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<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
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<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
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course/exam-02.html
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@@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter contains-active"> <li class="toc-item toc-chapter contains-active">
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@@ -135,7 +135,7 @@ eBookConfig.enable_chatcodes = false;
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course/frontmatter.html
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course/fun.html
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"body": " Suppose we have a coin that we're told is not fair and has a probability of of coming up heads. We suspect the probability is even higher than this. Throughout the following parts, do not use a normal approximation. We flip the coin 10 times and see 9 heads. Should we use a 1-tailed test or a 2-tailed test? Find the -value. Do we have strong enough evidence to reject the null hypothesis at a 0.05 significance level? Using a 1-tailed test, the -value is : The -value is the sum: , so we can reject the null hypothesis. What is the minimum number of heads we would need to see in 10 flips to reject the null hypothesis? We saw in part (a) that at least 9 heads is enough to reject. Then: Adding to the previous probabilities, . So 8 heads would not be enough to reject the null hypothesis, therefore the minimum number of heads to reject would be 9. What is the power of the test if the true value of the parameter is ? With : So the power of the test is . " "body": " Suppose we have a coin that we're told is not fair and has a probability of of coming up heads. We suspect the probability is even higher than this. Throughout the following parts, do not use a normal approximation. We flip the coin 10 times and see 9 heads. Should we use a 1-tailed test or a 2-tailed test? Find the -value. Do we have strong enough evidence to reject the null hypothesis at a 0.05 significance level? Using a 1-tailed test, the -value is : The -value is the sum: , so we can reject the null hypothesis. What is the minimum number of heads we would need to see in 10 flips to reject the null hypothesis? We saw in part (a) that at least 9 heads is enough to reject. Then: Adding to the previous probabilities, . So 8 heads would not be enough to reject the null hypothesis, therefore the minimum number of heads to reject would be 9. What is the power of the test if the true value of the parameter is ? With : So the power of the test is . "
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"body": " A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination. Sample Data best fit predicted res sum: The RSS is the sum of res , , and the SST is the sum of , . Then the coefficient of determination is: "
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course/notes-01-13.html
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course/notes-01-20.html
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course/notes-01-22.html
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course/notes-01-27.html
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course/notes-01-29.html
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course/notes-02-03.html
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course/notes-02-05.html
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course/notes-02-10.html
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<h1 class="heading"><a href="root-1-2.html"><span class="title">Introduction to Probability and Statistics for the Life Sciences</span> <span class="subtitle">Spring 2026 MATH 1044</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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<section class="worksheet" id="quiz-05" data-margins="0.5in 0.5in 0.5in 0.5in"><h2 class="heading hide-type">
<span class="type">Worksheet</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">Quiz 5</span><div class="print-links"><a href="?printpreview=quiz-05" class="print-link" title="Print preview"><span class="icon material-symbols-outlined" aria-hidden="true">&#xe8ad;</span></a></div>
</h2>
<section class="introduction" id="quiz-05-2"><div class="para" id="quiz-05-2-1">The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="para" id="quiz-05-2-2">Show all work unless instructed otherwise.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-2-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div></section><section class="onepage" id="quiz-05-3"><article class="exercise exercise-like" id="quiz-05-3-1"><h3 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="quiz-05-3-1-1-1">The joint and marginal distributions of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are given below. Find <span class="process-math">\(\Cov(X, Y)\)</span> and <span class="process-math">\(\rho_{X, Y}\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-3-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="quiz-05-3-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">127<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 Table<div class="autopermalink" data-description="Table 127: Joint Distribution Table"><a href="#quiz-05-3-1-1-2" title="Copy heading and permalink for Table 127: Joint Distribution Table" aria-label="Copy heading and permalink for Table 127: Joint Distribution Table">🔗</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 r0 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(Y = 2\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(Y = 3\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.1\)</span></td>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.15\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.2\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.3\)</span></td>
</tr>
</table></div></figure> <figure class="table table-like" id="quiz-05-3-1-1-3"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">128<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution Table for <span class="process-math">\(X\)</span><div class="autopermalink" data-description="Table 128: Distribution Table for X"><a href="#quiz-05-3-1-1-3" title="Copy heading and permalink for Table 128: Distribution Table for X" aria-label="Copy heading and permalink for Table 128: Distribution Table for X">🔗</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"><span class="process-math">\(x\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(0\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(X = x)\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.35\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.65\)</span></td>
</tr>
</table></div></figure> <figure class="table table-like" id="quiz-05-3-1-1-4"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">129<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Distribution Table for <span class="process-math">\(Y\)</span><div class="autopermalink" data-description="Table 129: Distribution Table for Y"><a href="#quiz-05-3-1-1-4" title="Copy heading and permalink for Table 129: Distribution Table for Y" aria-label="Copy heading and permalink for Table 129: Distribution Table for Y">🔗</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"><span class="process-math">\(y\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(2\)</span></th>
</tr>
<tr>
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(\Pr(Y = y)\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.25\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.35\)</span></td>
<td class="c m b0 r0 l0 t0 lines"><span class="process-math">\(0.4\)</span></td>
</tr>
</table></div></figure><div class="solutions"><details id="quiz-05-3-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-05-3-1-2-1">
<div class="para">
<span class="process-math">\(\Cov(X, Y) = \E(XY) - \E(X)\E(Y)\text{.}\)</span> We have:</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-2">
\begin{align*}
\E(X) \amp = 0(0.35) + 1(0.65) = 0.65 \\
\E(Y) \amp = 1(0.25) + 2(0.35) + 3(0.4) = 2.15 \\
\E(XY) \amp = (0)(1)(0.1) + (0)(2)(0.15) + (0)(3)(0.1) + \\
\amp \quad\, (1)(1)(0.15) + (1)(2)(0.2) + (1)(3)(0.3) = 1.45 \\ \\
\Cov(X, Y) \amp = 1.45 - (0.65)(2.15) = \boxed{0.0525}
\end{align*}
</div>
<div class="para">Since <span class="process-math">\(X\)</span> is an indicator random variable, we have <span class="process-math">\(\Var(X) = (0.65)(0.35) = 0.2275\text{.}\)</span> For <span class="process-math">\(\Var(Y)\text{:}\)</span>
</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-6">
\begin{align*}
\E(Y^2) \amp = 1^2(0.25) + 2^2(0.35) + 3^2(0.4) = 5.25 \\
\Var(Y) \amp = \E(Y^2) - (\E(Y))^2 = 5.25 - (2.15)^2 = 0.6275
\end{align*}
</div>
<div class="para">So the correlation is:</div>
<div class="displaymath process-math" id="quiz-05-3-1-2-1-7">
\begin{align*}
\rho_{X, Y} \amp = \frac{\Cov(X, Y)}{\sqrt{\Var(X)\Var(Y)}} = \frac{0.0525}{\sqrt{(0.2275)(0.6275)}} \approx \boxed{0.139}.
\end{align*}
</div>
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</div>
<div class="autopermalink" data-description="Solution 1.1"><a href="#quiz-05-3-1-2" title="Copy heading and permalink for Solution 1.1" aria-label="Copy heading and permalink for Solution 1.1">🔗</a></div>
</div></details></div>
<div class="autopermalink" data-description="Worksheet Exercise 1"><a href="#quiz-05-3-1" title="Copy heading and permalink for Worksheet Exercise 1" aria-label="Copy heading and permalink for Worksheet Exercise 1">🔗</a></div></article></section><section class="onepage" id="quiz-05-4"><article class="exercise exercise-like" id="quiz-05-4-1"><h3 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
<div class="para" id="quiz-05-4-1-1-1">A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column). Find the RSS, SST, and coefficient of determination.<div class="autopermalink" data-description="Paragraph"><a href="#quiz-05-4-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <figure class="table table-like" id="quiz-05-4-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">130<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Sample Data<div class="autopermalink" data-description="Table 130: Sample Data"><a href="#quiz-05-4-1-1-2" title="Copy heading and permalink for Table 130: Sample Data" aria-label="Copy heading and permalink for Table 130: Sample Data">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\(x_i\)</span></th>
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\(y_i\)</span></th>
<th scope="col" class="l m b1 r0 l0 t0 lines">best fit predicted <span class="process-math">\(y_i\)</span>
</th>
<th scope="col" class="l m b1 r0 l0 t0 lines">res<span class="process-math">\(^2\)</span>
</th>
<th scope="col" class="l m b1 r0 l0 t0 lines"><span class="process-math">\((y - \textrm{avg }y)^2\)</span></th>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(1.94\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(3.31\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(10.53\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(52.13\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(886.55\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(2.63\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(10.38\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(17.37\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(48.76\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(515.34\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(\vdots\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(3.94\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(19.47\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(30.31\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(117.35\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(185.24 \)</span></td>
</tr>
<tr>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(6.35\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(53.42\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(54.22\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(0.64\)</span></td>
<td class="l m b1 r0 l0 t0 lines"><span class="process-math">\(413.51\)</span></td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">sum:</td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(992.55\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(992.55\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(953.03\)</span></td>
<td class="l m b0 r0 l0 t0 lines"><span class="process-math">\(38397.35\)</span></td>
</tr>
</table></div></figure><div class="solutions"><details id="quiz-05-4-1-2" class="solution solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Solution</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="solution solution-like knowl__content">
<div class="para logical" id="quiz-05-4-1-2-1">
<div class="para">The RSS is the sum of res<span class="process-math">\(^2\text{,}\)</span> <span class="process-math">\(953.03\text{,}\)</span> and the SST is the sum of <span class="process-math">\((y - \textrm{avg})^2\text{,}\)</span> <span class="process-math">\(38397.35\text{.}\)</span> Then the coefficient of determination is:</div>
<div class="displaymath process-math" id="quiz-05-4-1-2-1-5">
\begin{gather*}
r^2 = 1 - \frac{\text{RSS}}{\text{SST}} = 1 - \frac{953.03}{38397.35} \approx \boxed{0.975}
\end{gather*}
</div>
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<span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">127</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span> <span class="type">Theorem</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">131</span><span class="period heading-divison-mark heading-divison-mark__period">.</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Fundamental Theorem of Calculus.</span>
</h3> </h3>
<div class="para logical" id="thm-FTC-2-1"> <div class="para logical" id="thm-FTC-2-1">
<div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div> <div class="para">If <span class="process-math">\(F(x)\)</span> is an antiderivative of <span class="process-math">\(f(x)\)</span>—i.e., if <span class="process-math">\(F'(x) = f(x)\)</span>—then:</div>
@@ -383,7 +384,7 @@ eBookConfig.enable_chatcodes = false;
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<article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3> <article class="exercise exercise-like" id="recitation-calculus-review-4-3"><h3 class="heading"><span class="codenumber">5<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h3>
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@@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">
course/syllabus.html
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@@ -252,6 +252,7 @@ eBookConfig.enable_chatcodes = false;
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-02.html" class="internal"><span class="title">Quiz 2</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-03.html" class="internal"><span class="title">Quiz 3</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li> <li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-04.html" class="internal"><span class="title">Quiz 4</span></a></div></li>
<li class="toc-item toc-worksheet"><div class="toc-title-box"><a href="quiz-05.html" class="internal"><span class="title">Quiz 5</span></a></div></li>
</ul> </ul>
</li> </li>
<li class="toc-item toc-chapter"> <li class="toc-item toc-chapter">