A buyer of household batteries wants to know the average lifetime of Duracell batteries. She takes a random sample of 55 Duracells. The Duracells have a sample lifetime mean of 72 hours and a sample standard deviation of 6 hours.

(a)Find a 90% confidence interval for the average lifetime of Duracell batteries.

(b)Find a 99% confidence interval for the average lifetime of Duracell batteries.

(c)Assume that the population standard deviation is equal to the sample standard deviation. What sample size would be needed to ensure that the buyer can estimate the average lifetime within 1.5 hours with 99% confidence?