Are younger New Zealand women marrying at a later age than previous generations of New Zealand women? To shed light on this question, we look at the proportions of New Zealand women who report having been married before age 22 for the two generations of New Zealand women.

(a) Out of the 314 women in the age group 2529,69 report having been married before the age 22. Find the posterior distribution of \({ }_{1}\), the proportion of New Zealand women in that age group who have married before age 22. (Use a uniform prior for \({ }_{1}\).)

(b) Out of the 219 women in the age group 50 54, 114 report having been married before age 22. Find the posterior distribution of \({ }_{2}\), the proportion of New Zealand women in that age group who have been married before age 22. (Use a uniform prior for \({ }_{2}\).)

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

(d) Test the hypothesis

\[
\begin{array}{lllllll}
H_{0}: & 1 & 2 & 0 & \text { versus } & H_{1}: & 1
\end{array} \quad 2<0
\]

in a Bayesian manner at the \(1 \%\) level of signi cance. Can we conclude that New Zealand women in the generation aged 2529 have married at an later age than New Zealand women in the generation aged 50 54 ?