Computers and Technology

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 < 0.80 (less than 80% have detectors), as opposed to p ≥ 0.80, the program will be implemented. Let x be the number of residences among the 25 that have a detector, and consider the following decision rule: Reject the claim that p = 0.8 and implement the program if x ≤ 15. (Round your answers to three decimal places.) (a) What is the probability that the program is implemented when p = 0.80? (b) What is the probability that the program is not implemented if p = 0.70? (c) What is the probability that the program is not implemented if p = 0.60? (d) How do the "error probabilities" of parts (b) and (c) change if the value 15 in the decision rule is changed to 14? The probabilities would decrease. The probabilities would increase. The probabilities would not change.

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