An ordinance requiring that a smoke detector beinstalled in all previously constructed houses has beenin effect in a particular city for 1 year. The fire departmentis concerned that many houses remain withoutdetectors. Let p 5 the true proportion of such houseshaving detectors, and suppose that a random sample of25 homes is inspected. If the sample strongly indicatesthat fewer than 80% of all houses have a detector, thefire department will campaign for a mandatory inspectionprogram. Because of the costliness of the program,the department prefers not to call for such inspectionsunless sample evidence strongly argues for their necessity.Let X denote the number of homes with detectorsamong the 25 sampled. Consider rejecting the claim thatp $ .8 if x # 15.a. What is the probability that the claim is rejectedwhen the actual value of p is .8?b. What is the probability of not rejecting the claimwhen p 5 .7? When p 5 .6?c. How do the “error probabilities” of parts (a) and (b)change if the value 15 in the decision rule is replacedby 14?
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 5 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
when the actual value of p is .8?
b. What is the probability of not rejecting the claim
when p 5 .7? When p 5 .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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