An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for 1 year. The fire department is concerned that many houses remain without detectors. Let p = the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than 80% of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let X denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that p ≥ .8 if x ≤ 15.
a. What is the probability that the claim is rejected when the actual value of p is .8?
b. What is the probability of not rejecting the claim when p = .7? when p = .6?
c. How do the “error probabilities” of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?
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