In a research program on human health risk from recreational contact with water contaminated with pathogenic microbiological material, the National Institute of Water and Atmospheric Research (NIWA) instituted a study to determine the quality of New Zealand stream water at a variety of catchment types. This study is documented in McBride et al. (2002), where \(n=116\) one-liter water samples from sites identi ed as having a heavy environmental impact from birds (seagulls) and waterfowl. Out of these samples, \(y=17\) samples 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 rst 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).