Are younger New Zealand women more likely to be in paid employment than previous generations of New Zealand women? To shed light on this question, we will look at the current employment status of two generations of New Zealand women 25 years apart.

(a) Out of 314 women in the age group 25-29, 171 were currently in paid employment. Find the posterior distribution of π1, the proportion of New Zealand women in that age group who are currently in paid employment.

(Use a uniform prior for π1.)

(b) Out of 219 women in the age group 50-54, 137 were currently in paid employment. Find the posterior distribution of π2, the proportion of New Zealand women in that age group who are currently in paid employment.

(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?