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## Section 5.3 Power of a Test

Text of section.
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### Exercises Exercises

#### 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, \(S\text{.}\) What values of \(S\) 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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#### 2.

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, \(S\text{.}\) What values of \(S\) 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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#### 3.

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.
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##### (a)

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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##### (b)

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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##### (c)

Suppose that this die is weighted so that it rolls a 1 with probability 0.2. What would be the power of our test?
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##### (d)

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.
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#### 4.

Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in\(^2\text{.}\) 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 \(p = 0.05\) 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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