Have the educational qualifications of younger New Zealand women changed from those of previous generations of New Zealand women? To shed light on this question, we will compare the educational qualifications of two generations of New Zealand women 25 years apart. The women in the age group 25-29 at the time of the survey were born between 1966 and 1970. The women in the age group 50-54 at the time of the survey were born between 1941 and 1945.

(a) Out of 314 women in the age group 25-29, 234 had completed a secondary school qualification. Find the posterior distribution of π1, the proportion of New Zealand women in that age group who have a completed a secondary school qualification. (Use a uniform prior for π1.)

(b) Out of 219 women in the age group 50-54, 120 had completed a secondary school qualification. Find the posterior distribution of π2, the proportion of New Zealand women in that age group who have a completed a secondary school qualification. (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?