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<div class="para" id="Exam-2-Review-4-14-1-1">We have an established factory which produces coins that are close to fair. We’re opening up a second factory, and we’d like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. We pick one sample coin from each factory, and flip each sample coin 100 times. The coin from the established factory flips 52 heads in 100 flips. The coin from the new factory flips 62 heads in 100 flips. Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-14-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 14"><a href="#Exam-2-Review-4-14" title="Copy heading and permalink for Exercise 14" aria-label="Copy heading and permalink for Exercise 14">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-15"><h4 class="heading"><span class="codenumber">15<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-2-Review-4-15-1-1">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-4-15-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div> <figure class="table table-like" id="Exam-2-Review-4-15-1-2"><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-4-15-1-2" 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">
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<div class="para" id="Exam-2-Review-4-15-1-1">Suppose we find a coin and wonder whether it’s fair. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-15-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 15"><a href="#Exam-2-Review-4-15" title="Copy heading and permalink for Exercise 15" aria-label="Copy heading and permalink for Exercise 15">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-16"><h4 class="heading"><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-2-Review-4-16-1-1">Suppose we have a coin which we suspect comes up heads more often than a fair coin would. As a first test, we decide to flip the coin 200 times and count the number of heads, <span class="process-math">\(S\text{.}\)</span> What values of <span class="process-math">\(S\)</span> would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-16-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Exercise 16"><a href="#Exam-2-Review-4-16" title="Copy heading and permalink for Exercise 16" aria-label="Copy heading and permalink for Exercise 16">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-17"><h4 class="heading"><span class="codenumber">17<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="introduction" id="Exam-2-Review-4-17-1"><div class="para" id="Exam-2-Review-4-17-1-1">Suppose we find a six-sided die and wonder whether it’s fair. As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. The expected number of 1’s is 50/3, with a variance of 125/9.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-17-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div></div>
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<article class="task exercise-like" id="Exam-2-Review-4-17-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
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<div class="para" id="Exam-2-Review-4-17-2-1-1">Using a normal approximation, what is the smallest number of 1’s greater than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-17-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 17.a"><a href="#Exam-2-Review-4-17-2" title="Copy heading and permalink for Task 17.a" aria-label="Copy heading and permalink for Task 17.a">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-17-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
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<div class="para" id="Exam-2-Review-4-17-3-1-1">Using a normal approximation, what is the greatest number of 1’s less than 50/3 that would be extreme enough to reject the null hypothesis of a fair die?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-17-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 17.b"><a href="#Exam-2-Review-4-17-3" title="Copy heading and permalink for Task 17.b" aria-label="Copy heading and permalink for Task 17.b">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-17-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
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<div class="para" id="Exam-2-Review-4-17-4-1-1">Suppose that this die is weighted so that it rolls a 1 with probability 0.2. What would be the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-17-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 17.c"><a href="#Exam-2-Review-4-17-4" title="Copy heading and permalink for Task 17.c" aria-label="Copy heading and permalink for Task 17.c">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-17-5"><h5 class="heading"><span class="codenumber">(d)</span></h5>
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<div class="para" id="Exam-2-Review-4-17-5-1-1">Suppose we roll the die 100 times and see 23 1’s. Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-17-5-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 17.d"><a href="#Exam-2-Review-4-17-5" title="Copy heading and permalink for Task 17.d" aria-label="Copy heading and permalink for Task 17.d">🔗</a></div></article><div class="autopermalink" data-description="Exercise 17"><a href="#Exam-2-Review-4-17" title="Copy heading and permalink for Exercise 17" aria-label="Copy heading and permalink for Exercise 17">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-18"><h4 class="heading"><span class="codenumber">18<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="para" id="Exam-2-Review-4-18-1-1">Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<span class="process-math">\(^2\text{.}\)</span> We suspect that growing this plant in a greenhouse will increase its height. We take the average height of a sample of 50 plants grown in a greenhouse. What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the <span class="process-math">\(p = 0.05\)</span> significance level? If the plants, when grown in a greenhouse, would truly have an average height of 41 in, what is the power of our test?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-18-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<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>
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<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>
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</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">
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<tr class="header-horizontal">
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<th scope="col" class="c m b1 r0 l0 t0 lines">drink type</th>
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<th scope="col" class="c m b1 r0 l0 t0 lines">water</th>
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@@ -496,11 +520,11 @@ eBookConfig.enable_chatcodes = false;
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<td class="c m b0 r0 l0 t0 lines">26</td>
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<td class="c m b0 r0 l0 t0 lines">14</td>
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</tr>
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</table></div></figure><div class="autopermalink" data-description="Exercise 15"><a href="#Exam-2-Review-4-15" title="Copy heading and permalink for Exercise 15" aria-label="Copy heading and permalink for Exercise 15">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-16"><h4 class="heading"><span class="codenumber">16<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="introduction" id="Exam-2-Review-4-16-1">
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<div class="para" id="Exam-2-Review-4-16-1-1">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-4-16-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</table></div></figure><div class="autopermalink" data-description="Exercise 19"><a href="#Exam-2-Review-4-19" title="Copy heading and permalink for Exercise 19" aria-label="Copy heading and permalink for Exercise 19">🔗</a></div></article><article class="exercise exercise-like" id="Exam-2-Review-4-20"><h4 class="heading"><span class="codenumber">20<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
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<div class="introduction" id="Exam-2-Review-4-20-1">
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<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>
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</div>
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<figure class="table table-like" id="Exam-2-Review-4-16-1-2"><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-4-16-1-2" 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">
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<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">
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<tr class="header-horizontal">
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<th scope="col" class="c m b1 r0 l0 t0 lines"># who respond to drug</th>
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<th scope="col" class="c m b1 r0 l0 t0 lines">0</th>
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@@ -519,16 +543,16 @@ eBookConfig.enable_chatcodes = false;
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</tr>
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</table></div></figure>
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<article class="task exercise-like" id="Exam-2-Review-4-16-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
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<div class="para" id="Exam-2-Review-4-16-2-1-1">Is the data consistent with a binomial distribution with parameter <span class="process-math">\(\theta = 0.7\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-16-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<article class="task exercise-like" id="Exam-2-Review-4-20-2"><h5 class="heading"><span class="codenumber">(a)</span></h5>
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<div class="para" id="Exam-2-Review-4-20-2-1-1">Is the data consistent with a binomial distribution with parameter <span class="process-math">\(\theta = 0.7\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-20-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 16.a"><a href="#Exam-2-Review-4-16-2" title="Copy heading and permalink for Task 16.a" aria-label="Copy heading and permalink for Task 16.a">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-16-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
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<div class="para" id="Exam-2-Review-4-16-3-1-1">What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation <span class="process-math">\(\widehat{\theta}\)</span> for the probability that a person will respond to the drug?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-16-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 20.a"><a href="#Exam-2-Review-4-20-2" title="Copy heading and permalink for Task 20.a" aria-label="Copy heading and permalink for Task 20.a">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-20-3"><h5 class="heading"><span class="codenumber">(b)</span></h5>
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<div class="para" id="Exam-2-Review-4-20-3-1-1">What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation <span class="process-math">\(\widehat{\theta}\)</span> for the probability that a person will respond to the drug?<div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-20-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 16.b"><a href="#Exam-2-Review-4-16-3" title="Copy heading and permalink for Task 16.b" aria-label="Copy heading and permalink for Task 16.b">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-16-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
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<div class="para" id="Exam-2-Review-4-16-4-1-1">Is the data consistent with a binomial distribution with the MLE value of <span class="process-math">\(\widehat{\theta}\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-16-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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<div class="autopermalink" data-description="Task 20.b"><a href="#Exam-2-Review-4-20-3" title="Copy heading and permalink for Task 20.b" aria-label="Copy heading and permalink for Task 20.b">🔗</a></div></article><article class="task exercise-like" id="Exam-2-Review-4-20-4"><h5 class="heading"><span class="codenumber">(c)</span></h5>
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<div class="para" id="Exam-2-Review-4-20-4-1-1">Is the data consistent with a binomial distribution with the MLE value of <span class="process-math">\(\widehat{\theta}\text{?}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#Exam-2-Review-4-20-4-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="autopermalink" data-description="Task 16.c"><a href="#Exam-2-Review-4-16-4" title="Copy heading and permalink for Task 16.c" aria-label="Copy heading and permalink for Task 16.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 16"><a href="#Exam-2-Review-4-16" title="Copy heading and permalink for Exercise 16" aria-label="Copy heading and permalink for Exercise 16">🔗</a></div></article><div class="autopermalink" data-description="Exercises"><a href="#Exam-2-Review-4" title="Copy heading and permalink for Exercises" aria-label="Copy heading and permalink for Exercises">🔗</a></div></section><div class="autopermalink" data-description="Section: Exam 2 Review"><a href="#Exam-2-Review" title="Copy heading and permalink for Section: Exam 2 Review" aria-label="Copy heading and permalink for Section: Exam 2 Review">🔗</a></div></section></div>
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<div class="autopermalink" data-description="Task 20.c"><a href="#Exam-2-Review-4-20-4" title="Copy heading and permalink for Task 20.c" aria-label="Copy heading and permalink for Task 20.c">🔗</a></div></article><div class="autopermalink" data-description="Exercise 20"><a href="#Exam-2-Review-4-20" title="Copy heading and permalink for Exercise 20" aria-label="Copy heading and permalink for Exercise 20">🔗</a></div></article><div class="autopermalink" data-description="Exercises"><a href="#Exam-2-Review-4" title="Copy heading and permalink for Exercises" aria-label="Copy heading and permalink for Exercises">🔗</a></div></section><div class="autopermalink" data-description="Section: Exam 2 Review"><a href="#Exam-2-Review" title="Copy heading and permalink for Section: Exam 2 Review" aria-label="Copy heading and permalink for Section: Exam 2 Review">🔗</a></div></section></div>
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