Are young New Zealand women having their children at a later age than previous generations?

(a) Out of the 314 women in the age group 25 29, 136 report having given birth to their rst child before the age of 25. Find the posterior distribution of 1 , the proportion of New Zealand women in that age group who have given birth before age 25. (Use a uniform prior for 1.)

(b) Out of the 219 women in the age group 50 54, 135 report having given birth to their rst child before age 25. Find the posterior distribution of 2 , the proportion of New Zealand women in that age group who have given birth before age 25. (Use a uniform prior for \({ }_{2}\).)

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

(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 had their rst child at a later age than New Zealand women in the generation aged 5054 ?