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 X be the number of policyholders in the sample who have smoke detectors.

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.