An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for one 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, where x is the observed value of X.
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 the actual value of p is .8?
c) how do the error probabilities of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?