Question: (10 pts) A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is 15 inches, and the standard deviation is 2 inches. A random sample of 36 fish is taken from the tank.

a.) State the random variable.

b.) What is the mean of the sample mean, µx?

c.) What is the standard deviation of the sample mean, σx?

d.) Draw the distribution for the sample mean, \(\bar{X}\).

e.) What is the probability that the sample mean is within 0.5 inches of the population mean?

f.) What is the probability that the sample mean is less than 14 inches?

g.) Suppose that you do pull out 36 fish from the tank and they do have a sample mean less than 14 inches, what would you think about the hatchery’s claim? Would you continue using this hatchery?