7.57 A city ordinance requires that a smoke detector be installed in all residential housing. There is concern that too many residences are still without detectors, so a costly inspection program is being contemplated.

Let p be the proportion of all residences that have a detector. A random sample of 25 residences is selected. If the sample strongly suggests that p , .80 (less than 80% have detectors), as opposed to p $ .80, the program will be implemented. Let x be the number of residences among the 25 that have a detector, and consider the following decision rule: Reject the claim that p $ .8 and implement the program if x # 15.

a. What is the probability that the program is implemented when p 5 .80?

b. What is the probability that the program is not implemented if p 5 .70?

c. What is the probability that the program is not implemented if p 5 .60?

d. How do the “error probabilities” of Parts

(b) and

(c) change if the value 15 in the decision rule is changed to 14?