The same study found that y = 12 out of n = 145 samples identified as having a heavy environmental impact from dairy farms contained Giardia cysts.

(a) What is the distribution of y, the number of samples containing Giardia cysts?

(b) Let π be the true probability that a one-liter water sample from this type of site contains Giardia cysts. Use a beta (1, 4) prior for π. Find the posterior distribution of π given y.

(c) Summarize the posterior distribution by its first two moments.

(d) Find the normal approximation to the posterior distribution g(π|y).

(e) Compute a 95% credible interval for π using the normal approximation in part (c).