1. A hat company states that the mean hat size for a male is at least 7.25. A random sample of 32 hat sizes has a mean of 7.15 and a standard deviation of 0.35.
   1. At = 0.05, can you reject the company's claim that the mean hat size for a male is at least 7.25? Be sure to state the p-value for this test.
   2. In the context of the problem, describe what a Type I error and a Type II error would mean.
2. The mean ACT score for 43 male high school students is 21.1 and the standard deviation is 5.0. The mean ACT score for 56 female high school students is 20.9 and the standard deviation is 4.7. At the 1% significance level, can you reject the claim that male and female high school students have equal ACT score averages? Be sure to use the p-value to make your conclusion.
3. A medical research team conducted a study to test the effect of a migraine drug. Of the 400 subjects who took the drug, 100 were pain free after two hours. Of the 407 subjects that took a placebo, 40 were pain free after two hours. Can you reject the claim that the proportion of subjects who are pain-free is the same for the two groups? Use the appropriate hypothesis test at the 2% significance level. Be sure to use the p-value to make your conclusion.
4. The Federal Bureau of Prisons publishes data in Prison Statisticson the times served by prisoners released from federal institutions for the first time. Independent random samples of released prisoners in the fraud and firearms offense categories yielded the following information on time served, in months. At the 5% significance level, do the data provide sufficient evidence to conclude that the mean time served for fraud is less than that for firearms offenses?

Dataset:

Fraud	Firearm
11.9	20.1
12.3	10.9
13.1	18.4
11.4	28.2
11.5	17.5
14	12.2
11.2	18.9
7.7	19.3
9.3	20.4
12.8	14.8