Problem 1
A country and western supply store would like to create a "one size fits most" cowboy hat for men. They would like to size this hat for the average man (in terms of head size) in order to have the hat fit "most" customers fairly well. It is known that male head size is normally distributed with a variance of 4cm2.
a) A SRS of 5 men is taken and their heads are measured to be 53cm, 50cm, 52cm, 51cm and 49cm. Determine a 99% confidence interval for mean head size based on these data using the following steps:
i. What is the distribution of the standardized sample mean? Why?
ii. Using the distribution from (i), compute a 99% confidence interval for mean head size in males. Interpret this interval in the context of the study.
b) The supply store will take another SRS of size n and create their hats based on the sample mean. They would like to be 99% confident that the true mean lies within 1 cm of the resulting sample mean. What is the smallest size n for which this will be true?
Problem 2
The box of Bob's favorite cereal states that the net contents weigh 12 ounces. Bob is suspicious because the package never seems full to him. He measures the weight of the next 20 boxes he buys and determines that the sample mean weight of the 20 boxes is 11.5 ounces. Assume cereal box weight is normally distributed with a standard deviation of 1.5 ounces.
a) Bob will assess the cereal company's claim that a box weighs 12 ounces based on the mean weight of a box of this cereal. What is the best alternative hypothesis for Bob's research question? What is the corresponding null hypothesis?
b) Bob (and you) will perform a 0.05 level hypothesis test using the following steps:
i. State the standardized test statistic under H0 and determine the distribution of this statistic. Why is this the correct distribution?
ii. Determine the realization of the test statistic based on the data collected by Bob.
iii. State the rejection region for this test. Should Bob reject his null hypothesis based on the rejection region? Give support for your answer.
iv. Determine the p-value of this test. Should Bob reject his null hypothesis based on the p-value? Give support for your answer.
Problem 3
Due to the relationship between heart disease and high cholesterol, it is recommended that individuals keep their cholesterol level below 200. A large insurance company wants to determine if its policyholders' cholesterol levels on average exceed the recommended amount. To obtain the data for this study, the insurance company randomly selects 50 policyholders from its database and offers them a free cholesterol screening. Assume cholesterol levels have a known standard deviation of 10 and that the type one error rate (α) for this study is set at 0.01.
a) What alternative hypothesis is appropriate for this study? What is the corresponding null hypothesis?
b) The sample mean of the cholesterol levels of the 50 policyholders is 204. Perform a hypothesis test using the following steps:
i. State the standardized test statistic under H0 and determine the distribution of this statistic. Why is this the correct distribution?
ii. Determine the realization of the test statistic based on the data collected by the insurance company.
iii. State the rejection region for this test. What is the conclusion of this test based on the rejection region in the context of the problem? Give support for your answer.
iv. Determine the p-value of this test. What is the conclusion of this test based on the p-value? Give support for your answer.
Problem 4
A door to door vaccuum sales company lets its salespeople set the price of the vaccuum he or she sells as long as the company makes a profit on the set price. An industry study indicates that on average the price that maximizes company profit is $125. The company would like to know if the average price that its salespeople are selling the vaccuum cleaners at is different than the target average price of $125. Twenty salespeople from this company are randomly selected and asked, "How much are you currently charging for the vaccuum cleaners?" The company knows the set price of vaccuum cleaners among the salespeople in the company is normally distributed with a variance of 225. They will use a type I error rate of 0.05.
a) What alternative hypothesis is appropriate for this study? What is the corresponding null hypothesis?
b) The sample mean of the set price of vaccuums for the 20 salespeople is $120. Perform a hypothesis test using the following steps:
i. State the standardized test statistic under H0 and determine the distribution of this statistic. Why is this the correct distribution?
ii. Determine the realization of the test statistic based on the data collected by the insurance company.
iii. State the rejection region for this test. What is the conclusion of this test based on the rejection region in the context of the problem? Give support for your answer.
iv. Determine the p-value of this test. What is the conclusion of this test based on the p-value? Give support for your answer.
c) For a two-sided α level test only, you can make a decision based on the RR, p-value, or a (1 - α)100% confidence interval. To see how to make a decision using a (1 - α)100% confidence interval, do the following:
i. Create a 95% confidence interval for the mean set price of vaccuum cleaners.
ii. If µ0 lies in the (1 -α)100% confidence interval for µ, we fail to reject the two-sided null hypothesis at level α. If µ0 lies outside this confidence interval, we reject the two-sided null hypothesis at level α. What decision is made for the hypothesis test described in (b) based on the confidence interval determined in (i)?
Note: the relationship between a α level two-sided test and a (1 - α)100% confidence interval not only holds for this problem, but is generally true.