Question: 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 P.M., 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 P.M. 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 P.M. 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 P.M. on that day. The computer output based on the data for 20 days provided the following results:
y^ = -432 + 7.7x, se = 28.17, SSxx = 607, and x¯ = 87.5
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 P.M.? 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 90F. 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.