Family formation patterns in New Zealand have changed over the time frame covered by the survey. New Zealand society has become more accepting of couples co-habiting (living together before or instead of legally marrying).

When we take this into account, are younger New Zealand women forming family like units at a similar age to previous generations?

(a) Out of the 314 women in the age group 25-29, 199 report having formed a domestic partnership (either co-habiting or legal marriage) before age 22. Find the posterior distribution of π1, the proportion of New Zealand women in that age group who have formed a domestic partnership before age 22. (Use a uniform prior for π1.)

(b) Out of the 219 women in the age group 50-54, 116 report having formed a domestic partnership before age 22. Find the posterior distribution of π2, the proportion of New Zealand women in that age group who have formed a domestic partnership before age 22. (Use a uniform prior for π2.)

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

(d) Find a 99% Bayesian credible interval for π1 − π2.

(e) What would be the conclusion if you tested the hypothesis H0 : π1 − π2 = 0 versus H1 : π1 − π2 = 0 at the 1% level of significance.