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Update to PreTeXt project source.

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andy
2026-04-02 14:27:33 -04:00
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source/review/Exam-2-Review.ptx
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@@ -28,7 +28,6 @@
</subsection> </subsection>
<exercises> <exercises>
<exercise> <exercise>
<statement> <statement>
<p> <p>
@@ -114,8 +113,7 @@
</task> </task>
</exercise> </exercise>
<exercise>
<exercise>
<statement> <statement>
<p> <p>
Suppose we flip a coin 100 times and count 60 heads. Suppose we flip a coin 100 times and count 60 heads.
@@ -273,7 +271,7 @@
</answer> </answer>
</exercise> </exercise>
<exercise> <exercise>
<introduction> <introduction>
<p> <p>
In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test. In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test.
@@ -360,7 +358,6 @@
</statement> </statement>
</exercise> </exercise>
<exercise> <exercise>
<statement> <statement>
<p> <p>
@@ -384,102 +381,178 @@
</statement> </statement>
</exercise> </exercise>
<exercise>
<statement>
<p>
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, <m>S</m>.
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?
</p>
</statement>
</exercise>
<exercise> <exercise>
<statement>
<p>
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, <m>S</m>.
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?
</p>
</statement>
</exercise>
<exercise>
<introduction>
<p>
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.
</p>
</introduction>
<task>
<statement> <statement>
<p> <p>
A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. 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?
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?
</p> </p>
<table>
<title></title>
<tabular>
<row header="yes" bottom="minor">
<cell halign="center">drink type</cell>
<cell halign="center">water</cell>
<cell halign="center">soda</cell>
<cell halign="center">tea</cell>
<cell halign="center">coffee</cell>
<cell halign="center">energy drinks</cell>
</row>
<row>
<cell halign="center">favorite</cell>
<cell halign="center">28</cell>
<cell halign="center">17</cell>
<cell halign="center">15</cell>
<cell halign="center">26</cell>
<cell halign="center">14</cell>
</row>
</tabular>
</table>
</statement> </statement>
</exercise> </task>
<exercise>
<introduction> <task>
<statement>
<p> <p>
A particular drug is administered in 100 independent trials. 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?
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.
</p> </p>
</statement>
<table> </task>
<title></title>
<tabular>
<row header="yes" bottom="minor">
<cell halign="center"># who respond to drug</cell>
<cell halign="center">0</cell>
<cell halign="center">1</cell>
<cell halign="center">2</cell>
<cell halign="center">3</cell>
<cell halign="center">4</cell>
</row>
<row>
<cell halign="center"># of trials</cell>
<cell halign="center">3</cell>
<cell halign="center">11</cell>
<cell halign="center">31</cell>
<cell halign="center">34</cell>
<cell halign="center">21</cell>
</row>
</tabular>
</table>
</introduction>
<task> <task>
<statement> <statement>
<p> <p>
Is the data consistent with a binomial distribution with parameter <m>\theta = 0.7</m>? Suppose that this die is weighted so that it rolls a 1 with probability 0.2.
</p> What would be the power of our test?
</statement> </p>
</task> </statement>
</task>
<task> <task>
<statement> <statement>
<p> <p>
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? Suppose we roll the die 100 times and see 23 1's.
</p> Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test.
</statement> </p>
</task> </statement>
</task>
</exercise>
<exercise>
<statement>
<p>
Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in<m>^2</m>.
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 <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?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
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?
</p>
<table>
<title></title>
<tabular>
<row header="yes" bottom="minor">
<cell halign="center">drink type</cell>
<cell halign="center">water</cell>
<cell halign="center">soda</cell>
<cell halign="center">tea</cell>
<cell halign="center">coffee</cell>
<cell halign="center">energy drinks</cell>
</row>
<row>
<cell halign="center">favorite</cell>
<cell halign="center">28</cell>
<cell halign="center">17</cell>
<cell halign="center">15</cell>
<cell halign="center">26</cell>
<cell halign="center">14</cell>
</row>
</tabular>
</table>
</statement>
</exercise>
<exercise>
<introduction>
<p>
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.
</p>
<table>
<title></title>
<tabular>
<row header="yes" bottom="minor">
<cell halign="center"># who respond to drug</cell>
<cell halign="center">0</cell>
<cell halign="center">1</cell>
<cell halign="center">2</cell>
<cell halign="center">3</cell>
<cell halign="center">4</cell>
</row>
<row>
<cell halign="center"># of trials</cell>
<cell halign="center">3</cell>
<cell halign="center">11</cell>
<cell halign="center">31</cell>
<cell halign="center">34</cell>
<cell halign="center">21</cell>
</row>
</tabular>
</table>
</introduction>
<task> <task>
<statement> <statement>
<p> <p>
Is the data consistent with a binomial distribution with the MLE value of <m>\widehat{\theta}</m>? Is the data consistent with a binomial distribution with parameter <m>\theta = 0.7</m>?
</p> </p>
</statement> </statement>
</task> </task>
</exercise>
<task>
<statement>
<p>
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?
</p>
</statement>
</task>
<task>
<statement>
<p>
Is the data consistent with a binomial distribution with the MLE value of <m>\widehat{\theta}</m>?
</p>
</statement>
</task>
</exercise>
</exercises> </exercises>
</section> </section>