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?

Solution : Step 1: It is given that the fire department conducted a survey find how many previously constructed houses have the smoke detector., if p=the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. And the sample shows that fewer than 80% of the hoses have the fire detector. The department claims that there is not to do inspections unless sample evidence strongly argues for their necessity. If X denote the number of homes with detectors among the 25 sampled. They rejecting the claim that p .8 if x 15.