# Introduction to Bayesian statistics - Bolstad M.

ISBN 0-471-27020-2

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230

BAYESIAN INFERENCE FOR DIFFERENCE BETWEEN MEANS

Edge of Pane Middle of Pane

1.51996 1.51997 1.52001 1.51999

1.51998 1.52000 1.52004 1.51997

1.51998 1.52004 1.52005 1.52000

1.52000 1.52001 1.52004 1.52002

1.52000 1.51997 1.52004 1.51996

For these data, y1 = 1.51999, y2 = 1.52001, a1 = .00002257, and

a2 = .00003075.

(a) Suppose glass at the edge of the pane is normal (p1 ,a2), where a1 = .00003. Calculate the posterior distribution of p1 when you use a normal (1.52000, .00012) prior for p1.

(b) Suppose glass in the middle of the pane is normal (p2,a2), where a2 = .00003. Calculate the posterior distribution of p2 when you use a normal (1.52000, .00012) prior for p2.

(c) Find the posterior distribution of pd = p1 - p2.

(d) Find a 95% credible interval for pd.

(e) Perform a Bayesian test of the hypothesis

H0 : pd = 0 versus pd = 0

at the 5% level of significance.

The last half of the twentieth century saw great change in the role of women in New Zealand society. These changes included education, employment, family formation, and fertility, where women took control of these aspects of their lives. During those years phrases such as "women’s liberation movement" and "the sexual revolution" were used to describe the changing role of women in society. In 1995 the Population Studies Centre at the University of Waikato sponsored the New Zealand Women Family, Employment, and Education Survey (NZFEE) to investigate these changes. A random sample of New Zealand women of all ages between 20 and 59 was taken, and the women were interviewed about their educational, employment, and personal history. The details of this survey are summarized in Marsaultet al. (1997). Detailed analysis of the data from this survey is in Johnstone et al. (2001).

12.8 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.

EXERCISES 231

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

(b) Out of 219 women in the age group 50-54,120 had completed a secondary school qualification. Find the posterior distribution of n2, the proportion of New Zealand women in that age group who have a completed a secondary school qualification. (Use a uniform prior for n2.)

(c) Find the approximate posterior distribution of nl — n2.

(d) Find a 99% Bayesian credible interval for nl — n2.

(e) What would be the conclusion if you tested the hypothesis

H0 : nl — n2 =0 versus Hl : nl — n2 =0 at the 1% level of significance?

12.9 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 nl, the proportion of New Zealand women in that age group who are currently in paid employment. (Use a uniform prior for nl.)

(b) Out of 219 women in the age group 50-54, 137 were currently in paid employment. Find the posterior distribution of n2, the proportion of New Zealand women in that age group who are currently in paid employment. (Use a uniform prior for n2.)

(c) Find the approximate posterior distribution of nl — n2.

(d) Find a 99% Bayesian credible interval for nl — n2.

(e) What would be the conclusion if you tested the hypothesis

H0 : nl — n2 =0 versus Hl : nl — n2 =0 at the 1% level of significance?

12.10 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 nl, the proportion

232

BAYESIAN INFERENCE FOR DIFFERENCE BETWEEN MEANS

of New Zealand women in that age group who had experienced sexual intercourse before age 18. (Use a uniform prior for ni.)

(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 n2, the proportion of New Zealand women in that age group who had experienced sexual intercourse before age 18. (Use a uniform prior for n2.)

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