An insurance company offers a discount to homeowners 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 Xbe the number of policyholders in the sample who have smoke detectors.

Question

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)?
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?
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.
d. If exactly 80% of the policyholders have smoke detectors, what is P(X ? 6)?
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?
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.

Answer 1

(a) X~Bin(8,0.80).





(b) Yes, if the claim is true, then only about 8 or 9 out of every 100,000 samples would have 1 or fewer policy holders with smoke detectors.



(c) Yes, because 1 out of 8 is an unusually small number of policy holders with smoke detectors if the claim is true.



(d)





(e) No, for nearly half the samples of size 8, 6 or fewer policy holders would have smoke detectors.



(f) No, 6 out of 8 is not an unusually small number if the claim is true.