In the same study of water quality, \(n=176\) samples were taken from streams having a high environmental impact from sheep farming. Out of these \(y=24\) 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

\[
H_{0}: \quad 15 \text { versus } H_{1}:<15
\]

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