Are younger New Zealand women more likely to be in paid employment than previous generations of New Zealand women? To shed light on this question, we will look at the current employment status of two generations of New Zealand women 25 years apart.

(a) Out of 314 women in the age group 2529,171 were currently in paid employment. Find the posterior distribution of \({ }_{1}\), the proportion of New Zealand women in that age group who are currently in paid employment. (Use a uniform prior for 1.)

(b) Out of 219 women in the age group 50 54, 137 were currently in paid employment. Find the posterior distribution of \({ }_{2}\), the proportion of New Zealand women in that age group who are currently in paid employment. (Use a uniform prior for 2 .)

(c) Find the approximate posterior distribution of \(1 \quad 2\).

(d) Find a \(99 \%\) Bayesian credible interval for \(1 \quad 2\).

(e) What would be the conclusion if you tested the hypothesis

\[
H_{0}:{ }_{1} \quad{ }_{2}=0 \text { versus } H_{1}:{ }_{1} \quad{ }_{2}=0
\]

at the \(1 \%\) level of significance?