8.6 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

(b) Let 7r be the true probability that a one-liter water sample from this type of site contains Giardia cysts. Use a beta(l,4) prior for 7r. Find the posterior distribution of 7r given y.

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

(d) Find the normal approximation to the posterior distribution g(7rIy).

(e) Compute a 95% credible interval for 7r using the normal approximation cysts?

found in part (d).