Chapter 4.2, Problem 20E

An insurance company offers a discount to home-owners who install smoke detectors in their homes. A company representative claims that 80% or more of policyholders have smoke detectors. You draw a random sample of eight policyholders. Let X be the number of policyholders in the sample who have smoke detectors.

1. a. If exactly 80% of the policyholders have smoke detectors (so the representative’s claim is true, but just barely), what is P(X £ 1)?
2. b. Based on the answer to part (a), if 80% of the policyholders have smoke detectors, would one policyholder with a smoke detector in a sample of size 8 be an unusually small number?
3. c. If you found that one of the eight sample policyholders had a smoke detector, would this be convincing evidence that the claim is false? Explain.
4. d. If exactly 80% of the policyholders have smoke detectors, what is P(X £ 6)?
5. e. Based on the answer to part (d), if 80% of the policyholders have smoke detectors, would six policyholders with smoke detectors in a sample of size 8 be an unusually small number?
6. f. If you found that six of the eight sample policyholders had smoke detectors, would this be convincing evidence that the claim is false? Explain.