Hoping to see through Lex's plan, Clark uses a fake name to place several orders for Megalopizzas, and he collects a random sample of 72 them for analysis. He measures the diameter of each pizza, enters the results in Microsoft Excel, and calculates several sample statistics, tabulated below.

Sample statistic	Value
Min	18.135 inches
Max	19.625 inches
Mean	18.740 inches
Median	18.755 inches
Standard deviation	0.370 inches

c. Using these sample statistics, calculate a 96% confidence interval for the true average diameter in the population of Megalopizzas.

d. Suppose that Clark is interested in suing Lex for false advertising, for which he would need to show strong evidence that Megalopizzas average less than 19 inches. Write down the null and alternative hypotheses for the hypothesis test that Clark should run. Which outcome of the hypothesis testrejection of the null hypothesis or failure to reject the null hypothesiswould constitute compelling evidence in support of Clark's false advertising case? DO NOT perform the test.

e. Clark believes that the average Megalopizza is approximately 18.65 inches in diameter and wishes to test whether he has compelling reason to doubt this belief. Using the sample statistics in the table, perform the following test at the 10% level of significance:

H0: = 18.65 vs. Ha: 18.65

Show all relevant calculations, draw a picture, and explain how you reached your conclusion. Give your conclusion in terms of the statement of the problem. What is the prob-value for the test?

A few days pass before Clark sees another of Lex's advertisements. This time, Lex's ad claims that "two-thirds of people prefer Lex's pizza to Clark's pizza."

"Only 33% of people prefer my pizza!?" Clark exclaims in disbelief. Livid once again, he vows to fight Lex with data. He hatches a plan to assemble his own focus group for a blind taste test.

f. What is the minimum number of randomly selected people Clark should include in his focus group if he wants to be sure that he will be able to calculate a 90% confidence interval no larger than "plus or minus 0.04" for the true proportion of people who prefer Clark's pizza?

g. Suppose it were true that the population proportion of people who prefer Clark's pizza is only 33%. In a focus group of 15 people, what is the probability that the number of people who prefer Clark's pizza is at least 4 and no more than 7?

h. Clark assembles a focus group of 50 people, and finds that 19 of them prefer Clark's pizza. Based on this result, perform a hypothesis test to see whether Clark has compelling evidence that more than one-third of people prefer his pizza to Lex's. That is, test

H0: = 0.333 vs. Ha: > 0.333

Show all relevant calculations, draw a picture, and explain how you reached your conclusion. Give your conclusion in terms of the statement of the problem. What is the prob-value for the test? Perform your test at the 5% significance level.