6.4 Suppose that you computed a 95% confidence interval for a population mean. The user of the statistics claims your interval is too wide to have any meaning in the specific use for which it is intended. Discuss and compare two methods for solving this problem.

6.12 Beechcraft, Inc., wants to estimate the average time it takes the Beechjet corporate jet to climb from sea level to 41,000 feet. From previous experience, company engineers believe that the standard deviation of climbing times is two minutes. The model is tested in 100 random trials, and the sample mean is found to be 22 minutes. Give a 95% confidence interval for the average climbing time from sea level to 41,000 feet.

6.16 Suppose you have a confidence interval based on a sample of size n. Using the same level of confidence, how large a sample is required to produce an interval of one-half the width?

6.18 A telephone company wants to estimate the average length of long-distance calls during weekends. A random sample of 50 calls gives a mean \(\overline{x}\) = 14.5 minutes and standard deviation s = 5.6 minutes. Give a 95% confidence interval and a 90% confidence interval for the average length of a long-distance phone calls during weekends.

6.24 An executive placement service needs to estimate the average salary of executives placed in a given industry. A random sample of 40 executives gives \(\overline{x}\) = $42,539 and s = $11,690. Give a 90% confidence interval for the average salary of an executive placed in this industry.

6.60 How many test runs of an automobile are required for determining its average miles-per-gallon rating on the highway to within 2 miles per gallon with 95% confidence, if a guess is that the variance of the population of miles per gallon is about 100?

6.64 Find the minimum required sample size for estimating the average number of designer shirts sold per day to within 10 units with 90% confidence if the standard deviation of the number of shirts sold per day is about 50.

6.44 A recent article describes the success of business schools in Europe and the demand on that continent for the MBA degree. The article reports that a survey of 280 European business positions resulted in the conclusion that only one-seventh of the positions for MBAs at European businesses are currently filled. Assuming that these numbers are exact and that the sample was randomly chosen from the entire population of interest, give a 90% confidence interval for the proportion of filled MBA positions in Europe.

7.2 A medicine is effective only if the concentration of a certain chemical in it is at least 200 parts per million (ppm). At the same time, the medicine would produce an undesirable side effect if the concentration of the same chemical exceeds 200 ppm. How would you set up the null and alternative hypothesis to test the concentration of the chemical in the medicine?

7.4 The average cost of a traditional open-heart surgery is claimed to be $49,160. If you suspect that the claim exaggerates the cost, how would you set up the null and alternative hypothesis?

1. How can the power of a hypothesis test be increased without increasing the sample size?
2. When planning a hypothesis test, what should be done if the probabilities of both type I and type II errors to be small?

7.12 The calculated z for a hypothesis test is -1.75. What is the p -value if the test is ( a ) left-tailed, ( b ) right-tailed, and ( c ) two-tailed?

7.16 An automobile manufacturer substitutes a different engine in cars that were known to have an average miles-per-gallon rating of 31.5 on the highway. The manufacturer wants to test whether the new engine changes the miles-per-gallon rating of the automobile model. A random sample of 100 trail runs gives \(\overline{x}\) = 29.8 miles per gallon and s = 6.6 miles per gallon. Using the 0.05 level of significance , is the average miles-per-gallon rating on the highway for cars using the new engine different from the rating for cars using the old engine?

1. An investment services company claims that the average annual return on stocks within a certain industry is 11.5%. An investor wants to test whether this claim is true and collects a random sample of 50 stocks in the industry of interest. He finds that the sample average annual return is 10.8% and the sample standard deviation is 3.4%. Does the investor have enough evidence to reject the investment company’s claim? (Use \(\alpha \) = 0.05).

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