A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of 40 hours and a standard deviation of 5 hours. Let X represent the mean lifetime of the batteries in a simple random sample of size 100.

a. If the claim is true, what is P(X ≤ 36.7)?

b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 36.7 hours unusually short?

c. If the sample mean lifetime of the 100 batteries were 36.7 hours, would you find the manufacturer’s claim to be plausible? Explain.

d. If the claim is true, what is P(X ≤ 39.8)?

e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short?

f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer’s claim to be plausible? Explain.