In the study of water quality in New Zealand streams, documented in McBride et al. (2002), a high level of Campylobacter was de ned as a level greater than 100 per \(100 \mathrm{ml}\) 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 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

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

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

(d) Find a \(95 \%\) credible interval for .

(e) Test the hypothesis

\[
H_{0}:=10 \text { versus } H_{1}:=10
\]

at the \(5 \%\) level of signi cance.