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 77oF to 95oF 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:
ŷ = –432 + 7.7x, se = 28.17, SSxx = 607, and = 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 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 90o. Answer using a 95% confidence level.
c. The owner has seen tomorrow’s weather forecast, which predicts a high of 90oF. 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 100oF. Give a point estimate, together with an appropriate warning to the owner.