The same study found that \(y=10\) out of \(n=174\) samples identified as having a heavy environmental impact from pastoral (sheep) farms contained Giardia cysts.

(a) What is the distribution of \(y\), the number of samples containing \(G i\) ardia cysts?

(b) Let 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 . Find the posterior distribution of given \(y\).

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

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

(e) Compute a 95\% credible interval for using the normal approximation found in part (d).