Are younger New Zealand women becoming sexually active at an earlier 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 experienced sexual intercourse before age 18 for the two generations of New Zealand women.

(a) Out of the 298 women in the age group 2529 who responded to this question, 180 report having experienced sexual intercourse before reaching the age of 18 . Find the posterior distribution of \({ }_{1}\), the proportion of New Zealand women in that age group who had experienced sexual intercourse before age 18. (Use a uniform prior for 1.)

(b) Out of the 218 women in the age group 5054 who responded to this question, 52 report having experienced sexual intercourse before reaching the age of 18 . Find the posterior distribution of \({ }_{2}\), the proportion of New Zealand women in that age group who had experienced sexual intercourse before age 18. (Use a uniform prior for \({ }_{2}\).)

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

(d) Test the hypothesis

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

in a Bayesian manner at the \(1 \%\) level of signi cance. Can we conclude that New Zealand women in the generation aged 2529 have experienced sexual intercourse at an earlier age than New Zealand women in the generation aged 5054 ?