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 π be the proportion of all residences that have a detector. A random sample of 25 residences is selected. If the sample strongly suggests that π < .80 (less than 80% have detectors), as opposed to π ≥ .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 π = .8 and implement the program if x ≤ 15.

a. What is the probability that the program is implemented when π = .80?

b. What is the probability that the program is not implemented if π = .70? if π = .60?

c. How do the “error probabilities” of Parts (a) and (b) change if the value 15 in the decision rule is changed to 14?