Suppose that you work part-time at a bowling alley that is open daily from noon to midnight. Although business is usually slow from noon to 6 PM, the owner has noticed that it is better on hotter days during the summer, perhaps because the premises are comfortably air-conditioned. The owner shows you some data that she gathered last summer. This data set includes the maximum temperature and the number of lines bowled between noon and 6 PM for each of 20 days. (The maximum temperatures ranged from 77° F to 95° F during this period.) The owner would like to know if she can estimate tomorrow's business from noon to 6 PM by looking at tomorrow's weather forecast. She asks you to analyze the data. Let x be the maximum temperature for a day and y the number of lines bowled between noon and 6 PM on that day. The computer output based on the data for 20 days provided the following results:
Assume that the weather forecasts are reasonably accurate.
a. Does the maximum temperature seem to be a useful predictor of bowling activity between noon and 6 PM? Use an appropriate statistical procedure based on the information given. Use α=.05.
b. The owner wants to know how many lines of bowling she can expect, on average, for days with a maximum temperature of 90 ?. Answer using a 95% confidence level.
c. The owner has seen tomorrow's weather forecast, which predicts a high of 90° F. About how many lines of bowling can she expect? Answer using a 95% confidence level.
d. Give a brief commonsense explanation to the owner for the difference in the interval estimates of parts b and c.
e. The owner asks you how many lines of bowling she could expect if the high temperature were 100° F. Give a point estimate, together with an appropriate warning to the owner.