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 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 25-29 who responded to this question, 180 report having experienced sexual intercourse before reaching the age of 18. Find the posterior distribution of π1, the proportion of New Zealand women in that age group who had experienced sexual intercourse before age 18. (Use a uniform prior for π1.)

(b) Out of the 218 women in the age group 50-54 who responded to this question, 52 report having experienced sexual intercourse before reaching the age of 18. Find the posterior distribution of π2, the proportion of New Zealand women in that age group who had experienced sexual intercourse before age 18. (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 experienced sexual intercourse at an earlier age than New Zealand women in the generation aged 50-54?