Chapter 5.12, Problem 7E

Refer to Exercise 5.5. Because the company is sampling the coffee production process every hour, there are 720 confidence intervals for the mean caffeine content μ constructed every month.

1. a. If the level of confidence remains at 95% for the 720 confidence intervals in a given month, how many of the confidence intervals would you expect to fail to contain the value of μ and hence provide an incorrect estimation of the mean caffeine content?
2. b. If the number of samples is increased from 50 to 100 each hour, how many of the 95% confidence intervals would you expect to fail to contain the value of μ in a given month?
3. c. If the number of samples remains at 50 each hour but the level of confidence is increased from 95% to 99% for each of the intervals, how many of the 99% confidence intervals would you expect to fail to contain the value of μ in a given month?

5.5 A company that manufacturers coffee for use in commercial machines monitors the caffeine content in its coffee. The company selects 50 samples of coffee every hour from its production line and determines the caffeine content. From historical data, the caffeine content (in milligrams, mg) is known to have a normal distribution with $\hat{\sigma }$ = 7.1 m During a 1-hour time period, the 50 samples yielded a mean caffeine content of $\bar{y}$ = 110 mg.

1. a. Identify the population about which inferences can be made from the sample data.
2. b. Calculate a 95% confidence interval for the mean caffeine content μ of the coffee produced during the hour in which the 50 samples were selected.
3. c. Explain to the CEO of the company in nonstatistical language the interpretation of the constructed confidence interval.