A manufacturer of automobile batteries claims that the distribution of the life of its best battery has a mean of 54 months and a standard deviation of 6 months. Suppose a consumer group decides to check the claim by purchasing a sample of 50 of these batteries and subjecting them to tests that determine battery life.

(a) Assuming the manufacturer's claim is true, describe the sampling distribution of the mean lifetime of a sample of 50 batteries.

(b) Assuming the manufacturer's claim is true, what is the probability the consumer group's sample has a mean life of 52 or fewer months?

(c) What would your results be if the consumer group used a sample of 35 batteries instead of 50? Explain your results.

The amount of time a bank teller spends with each customer has a population mean μ = 3.10 minutes and standard deviation σ = 0.40 minutes. If a random sample of 16 customers is selected,

(a) What is the probability that the average time spent per customer will be at least 3 minutes?
(b) There is an 85% chance that the sample mean will be below how many minutes?
(c) What assumption must be made in order to solve (a) and (b).
(d) If a random sample of 64 customers is selected, there is an 85% chance that the sample mean will be below how many minutes?
(e) What assumptions if any, must be made in order to solve (d)?
(f) Which is more likely to occur - an individual service time below two minutes, a sample mean above 3.4 minutes in a sample of 16 customers, or a sample mean below 2.9 minutes in a sample of 64 customers? Explain.