A manufacturer of automobile batteries claims that the distribution of the lengths of life of its best battery has a mean of 54 months and a standard deviation of 6 months. Recently, the manufacturer has received a rash of complaints from unsatisfied customers whose batteries have died earlier than expected. Suppose a consumer group decides to check the manufacturer's claim by purchasing a sample of 50 of these batteries and subjecting them to tests that determine battery life.

a. Assuming that the manufacturer's claim is true, what is the probability that the consumer group's sample has a mean life greater than 52.5 miles but less than 56 miles?

b. Find the value of the sample mean that divides the upper 1% of the sampling distribution of the sample mean from the lower 99%.