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Sectioning Ch 5

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<section xml:id="sec-Power" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Power of a Test</title>
<p>
Text of section.
</p>
<exercises xml:id="exercises-Power">
<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>
<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>
<p>
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?
</p>
</statement>
</task>
<task>
<statement>
<p>
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?
</p>
</statement>
</task>
<task>
<statement>
<p>
Suppose that this die is weighted so that it rolls a 1 with probability 0.2.
What would be the power of our test?
</p>
</statement>
</task>
<task>
<statement>
<p>
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.
</p>
</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>
</exercises>
</section>