In the study of water quality in New Zealand streams documented in McBride et al. (2002) a high level of Campylobacter was defined as a level greater than 100 per 100 ml of stream water. n = 116 samples were taken from streams having a high environmental impact from birds. Out of these y = 11 had a high Campylobacter level. Let π be the true probability that a sample of water from this type of stream has a high Campylobacter level.

(a) Find the frequentist estimator for π.

(b) Use a beta (1, 10) prior for π. Calculate the posterior distribution g(π|y).

(c) Find the posterior mean and variance. What is the Bayesian estimator for

π?

(d) Find a 95% Credible interval for π.

(e) Test the hypothesis H0 : π = .10 versus H1 : π = .10 at the 5% level of significance.