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 fol- lowing 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 = .80?
b. What is the probability that the program is not im- plemented if p = .70? if p = .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?