In the same study of water quality, \(n=87\) samples were taken from streams in municipal catchments. Out of these \(y=8\) 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) Test the hypothesis

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

at the \(5 \%\) level of significance.