A farmer is interested in knowing the mean weight of his chickens when they leave the farm. Suppose that the standard deviation of the chicken's weight is 500 grams.

a) What is the minimum number of chickens needed to ensure that the standard deviation of the sample mean is no more than 90 grams?

(b) Suppose the farm has three coops. The mean weights in each coop are 1.75, 1.85 and 2.1 kg, and standard deviations are 450, 520, and 380 grams, respectively. Calculate the probability that a random sample of 30 chickens from the first coop will have a mean weight larger than 1.925 kg. Calculate the same probability for the second and third coops.

(c) Suppose the proportion of the three coops are 0.60, 0.25, 0.15. Given that a random sample of 30 chickens from some coop has a mean weight larger than 1.925 kg, find the posterior probability the sample is from the (i) first coop, (ii) second coop, (iii) third coop. Which coop did the sample of chickens most likely have come from?