An insurance company offers a discount to homeowners who install smoke detectors in their homes. A company representative claims that 82% 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 somke detectors, would one policyholder with a smoke detectors in a sample of size 8 be an unusually small number?

c. If you found that exactly 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 exactly six of the eight sample policyholders had smoke detectors, would this be convincing evidence that the claim is false? Explain.

This variable is discrete, has only two possible outcomes "success": the policyholder has a smoke detector and "failure": the policyholder doesn't have a smoke detector. All outcomes are independent, the size of the experiment is fixed (n=8) and the probability of success is known, so we can conclude that this variable has a binomial distribution. I'll use a table of the binomial distribution to calculate the asked probabilities.

a. The probability that the policyholders have smoke detectors is exactly 80%

p=0.8

n=8

You have to calculate the probability that at most 1 policyholder has a smoke detector:

P(X≤1)= 0.00008448

b. Yes, the probability of finding a policyholder that has a smoke detector in a sample this size is extremely small. With such a low probability it is more likely that none of the policyholders sampled possess smoke detectors.

c. Not exactly, the probability of finding one "success" is very small but is not cero. The sample was taken at random from a bigger population where many policyholders have smoke detectors and there is a chance that at least one was chosen.

Remember, the probability of an impossible event is cero but an event with probability cero is not necessarily impossible.

d. Now you have to calculate the probability of at most 6 policyholders to have smoke detectors.

P(X≤6)= 0.49668352

e. The probability of finding at most 6 policyholders that have smoke detectors in the sample is almost 50%, this is neither an unusual number nor a small value.

The expected value for this example is E(X)= n*p= 8*0.8= 6.4, given the probability of success of 80% we expect that 6.4 policyholders have smoke detectors, i.e. the probability calculated in d. is an expected probability.

f. No, remember there is a difference between finding and expecting, you'd expect that 6 policyholders have smoke detectors, and this can occur with a certain probability (the higher it is the more chances its got to happen) but since samples are taken at random, it could happen that from the sample only two had smoke detectors, all of them or none. Finding 6 persons with smoke detectors will confirm whats expected but it isn't proof that every time you take a sample of 8, 6 of them will have them.

I hope it helps!