558 lines
16 KiB
XML
558 lines
16 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<section xml:id="Exam-2-Review">
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<title>Exam 2 Review</title>
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<introduction>
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<p>
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Use the following problems to prepare for the exam.
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There will be in-class review on Tuesday 3/31 and Thursday 4/2.
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Your recitation this week will also be exam review.
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</p>
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</introduction>
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<subsection>
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<title>Allowed Materials</title>
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<p>
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You will be allowed to use a scientific calculator (<em>not</em> a graphing calculator, <em>not</em> a calculator app on your phone).
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You may not share a calculator with another student; you must use your own calculator.
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</p>
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<p>
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You may bring a standard 3 in x 5 in index card with prepared notes.
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You may use both sides of the notecard.
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You must put your full name in the top right corner of the card, and turn it in along with your exam.
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</p>
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</subsection>
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<exercises>
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<exercise>
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<statement>
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<p>
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Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times.
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Let <m>S</m> be the number of heads.
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Estimate the probability that <m>34 \leq S \leq 44</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294.</m>
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</p>
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</answer>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose a fair die is rolled 100 times, and let <m>m</m> be the average value of the rolls.
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Estimate the probability that <m>3.45 \leq m \leq 3.55</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510.</m>
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</p>
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</answer>
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</exercise>
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<exercise>
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<introduction>
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<p>
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The heights of men in the US have a mean of 69 in and a variance of about 9 in<m>^2</m>, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in<m>^2</m>.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Suppose the heights of 30 men are sampled, and a sample mean <m>m</m> is taken.
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Find the expected value and variance of <m>m</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\E(m) = 69</m> and <m>\Var(m) = 0.3</m>.
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</p>
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</answer>
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</task>
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<task>
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<statement>
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<p>
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Estimate the probability that <m>m \geq 69.5</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814</m>.
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</p>
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</answer>
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</task>
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<task>
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<statement>
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<p>
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If the sampled group was women, estimate the probability that <m>m \geq 64</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>\E(m) = 63.5</m> and <m>\Var(m) \approx 0.208</m>. <m>\Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357</m>.
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</p>
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</answer>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we flip a coin 100 times and count 60 heads.
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Let <m>p</m> be the (unknown) probability that the coin comes up heads on a flip.
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Give an approximate 95% confidence interval for the value of <m>p</m>.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.504, 0.696]</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose in a sample of 100 people, 12 are left-handed.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Give a 95% confidence interval for the proportion <m>p</m> of left-handed people.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.056, 0.184]</m>.
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</p>
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</answer>
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</task>
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<task>
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<statement>
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<p>
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Give a 90% confidence interval.
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</p>
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</statement>
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<answer>
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<p>
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<m>[0.066, 0.174]</m>.
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</p>
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</answer>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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The weights of five mice are measured and recorded below.
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Give a 95% confidence interval for the sample mean weight of mice.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell>mouse <m>i</m></cell>
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<cell>weight <m>W_i</m> (g)</cell>
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</row>
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<row>
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<cell>1</cell>
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<cell>26</cell>
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</row>
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<row>
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<cell>2</cell>
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<cell>32</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>33</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>20</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>29</cell>
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</row>
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</tabular>
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</table>
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</statement>
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<answer>
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<p>
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<!-- avg weight = 28 --> <!-- sum of squares = 4030 --> <!-- s^2 = 27.5 --> <!-- \sqrt{s^2/n} \approx 2.35 --> <m>[23.39, 32.61]</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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<statement>
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<p>
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The heights of five plants are measured and recorded below.
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Give a 95% confidence interval around the sample mean for the heights of the plants.
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(Pretend 5 measurements is large enough for the CLT to apply.)
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</p>
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<table>
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<title></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell>plant <m>i</m></cell>
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<cell>height <m>H_i</m> (in)</cell>
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</row>
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<row>
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<cell>1</cell>
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<cell>15</cell>
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</row>
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<row>
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<cell>2</cell>
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<cell>14</cell>
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</row>
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<row>
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<cell>3</cell>
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<cell>18</cell>
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</row>
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<row>
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<cell>4</cell>
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<cell>21</cell>
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</row>
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<row>
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<cell>5</cell>
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<cell>17</cell>
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</row>
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</tabular>
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</table>
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</statement>
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<answer>
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<p>
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<!-- avg weight = 17 --> <!-- sum of squares = 1475 --> <!-- s^2 = 7.5 --> <!-- \sqrt{s^2/n} \approx 1.22 --> <m>[14.61, 19.39]</m>.
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</p>
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</answer>
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</exercise>
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<exercise>
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<introduction>
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<p>
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In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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We have a coin which we've flipped many times, seeing an above-average number of heads.
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We suspect the coin comes up heads more often than a fair coin would.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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We find a coin on the street and wonder whether or not it's a fair coin.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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We suspect there will be a difference in average weight of mice caught during the summer versus during the winter.
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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In a medical study, a group of patients are gathered and the proportion experiencing particular symptoms is measured.
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A new drug intended to eliminate these symptoms is administered, after which the proportion experiencing symptoms is measured again.
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</p>
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</statement>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we find a coin and wonder whether it's fair.
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As a first test, we decide to flip the coin 200 times and count the number of heads.
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If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we have a coin which we suspect comes up heads more often than a fair coin would.
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As a first test, we decide to flip the coin 200 times and count the number of heads.
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If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we find a six-sided die and wonder whether it's fair.
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As a first test, we decide to roll the die 100 times and count the number of times it comes up 1.
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If we roll 22 1's, should we accept or reject the null hypothesis of a fair die at a significance level of 0.05?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<m>^2</m>.
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We suspect that growing this plant in a greenhouse will increase its height.
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A sample of 50 plants grown in a greenhouse has an average height of 40 in.
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Is this significant enough data to reject the null hypothesis of equal means at the <m>p = 0.05</m> significance level?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety.
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A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams<m>^2</m> (for an individual apple).
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An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams<m>^2</m>.
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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We have an established factory which produces coins that are close to fair.
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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.
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We pick one sample coin from each factory, and flip each sample coin 100 times.
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The coin from the established factory flips 52 heads in 100 flips.
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The coin from the new factory flips 62 heads in 100 flips.
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Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we find a coin and wonder whether it's fair.
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As a first test, we decide to flip the coin 200 times and count the number of heads, <m>S</m>.
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What values of <m>S</m> 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?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose we have a coin which we suspect comes up heads more often than a fair coin would.
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As a first test, we decide to flip the coin 200 times and count the number of heads, <m>S</m>.
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What values of <m>S</m> 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?
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</p>
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</statement>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Suppose we find a six-sided die and wonder whether it's fair.
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As a first test, we decide to roll the die 100 times and count the number of times it comes up 1.
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The expected number of 1's is 50/3, with a variance of 125/9.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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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?
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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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?
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Suppose that this die is weighted so that it rolls a 1 with probability 0.2.
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What would be the power of our test?
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Suppose we roll the die 100 times and see 23 1's.
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Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.
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</p>
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</statement>
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</task>
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</exercise>
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<exercise>
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<statement>
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<p>
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Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<m>^2</m>.
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We suspect that growing this plant in a greenhouse will increase its height.
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We take the average height of a sample of 50 plants grown in a greenhouse.
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What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the <m>p = 0.05</m> 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?
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</p>
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</statement>
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</exercise>
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<exercise>
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<statement>
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<p>
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A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell.
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They survey their customers about their favorite drinks.
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Is the data below consistent with the null hypothesis that each type of drink will be equally preferred?
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</p>
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<table>
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<title></title>
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<tabular>
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<row header="yes" bottom="minor">
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<cell halign="center">drink type</cell>
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<cell halign="center">water</cell>
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<cell halign="center">soda</cell>
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<cell halign="center">tea</cell>
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<cell halign="center">coffee</cell>
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<cell halign="center">energy drinks</cell>
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</row>
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<row>
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<cell halign="center">favorite</cell>
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<cell halign="center">28</cell>
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<cell halign="center">17</cell>
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<cell halign="center">15</cell>
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<cell halign="center">26</cell>
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<cell halign="center">14</cell>
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</row>
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</tabular>
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</table>
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</statement>
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</exercise>
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<exercise>
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<introduction>
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<p>
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A particular drug is administered in 100 independent trials.
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In each trial, the drug is administered to four people, and we count how many respond to the drug.
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The table below shows how many trials have each different count of people who respond to the drug.
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</p>
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<table>
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<title></title>
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<tabular>
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<row header="yes" bottom="minor">
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<cell halign="center"># who respond to drug</cell>
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<cell halign="center">0</cell>
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<cell halign="center">1</cell>
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<cell halign="center">2</cell>
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<cell halign="center">3</cell>
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<cell halign="center">4</cell>
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</row>
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<row>
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<cell halign="center"># of trials</cell>
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<cell halign="center">3</cell>
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<cell halign="center">11</cell>
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<cell halign="center">31</cell>
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<cell halign="center">34</cell>
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<cell halign="center">21</cell>
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</row>
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</tabular>
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</table>
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</introduction>
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<task>
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<statement>
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<p>
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Is the data consistent with a binomial distribution with parameter <m>\theta = 0.7</m>?
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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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 <m>\widehat{\theta}</m> for the probability that a person will respond to the drug?
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</p>
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</statement>
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</task>
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<task>
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<statement>
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<p>
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Is the data consistent with a binomial distribution with the MLE value of <m>\widehat{\theta}</m>?
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</p>
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</statement>
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</task>
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</exercise>
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</exercises>
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</section> |