b. Calculate \(\operatorname{E}[X]\)

c. Calculate \(\operatorname{Var}[X]\)

2. Ahmadi, Inc., has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 64 of the automobiles. The sample showed an average of 51.5 miles per gallon with a standard deviation of 4 miles per gallon. Assume the population is normally distributed.(12%) a. Formulate the hypotheses to determine whether or not the manufacturer's advertising campaign is legitimate.

b. Compute the test statistic.

c. What is the \(p\)-value associated with the sample results and what is your conclusion? Let \(\alpha = .05\)

3. The Federal Bureau of Labor Statistics surveyed 50,000 people and found the unemployment rate to be \(5.8\%\). The margin of error was \(0.2\%\). Construct a confidence interval for the unemployment rate. What is the corresponding confidence level? Please interpret this confidence interval by words and graphs, and the confidence coefficient as well.(10%)

4. Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05 inches.(12%) a. What hypotheses should be tested, and why?

b. In this test, what would be a Type I error?

c. In this test, what would be a Type II error?

5. A beer producer claims that the proportion of its customers who cannot distinguish regular from light beer is at most 12 percent. The producer decides to test this null hypothesis against the alternative that the true proportion is more than 12 percent. The decision rule adopted is to reject the null hypothesis if the sample proportion of who cannot distinguish between the two beers exceeds 15 percent.(12%) a. If a random sample of 100 customers is chosen, what is the probability of a Type I error, using this decision rule?

b. Suppose that the true proportion of customers who cannot distinguish between these flavors is 0.20. If a random sample of 100 customers is selected, what is the probability of a Type II error?

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