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 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 25-29, 69 report having been married before the age 22. Find the posterior distribution of π1, the proportion of New Zealand women in that age group who have married before age 22. (Use a uniform prior for π1.)

(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 π2, the proportion of New Zealand women in that age group who have been married before age 22. (Use a uniform prior for π2.)

(c) Find the approximate posterior distribution of π1 − π2.

(d) Test the hypothesis H :0: π1 − π2 ≥ 0 versus H1 : π1 − π2 < 0 in a Bayesian manner at the 1% level of significance. Can we conclude that New Zealand women in the generation aged 25-29 have married at an earlier age than New Zealand women in the generation aged 50-54?