Previous research has suggested that the childhood circumcision of males may be a protective factor against the acquisition of sexually transmitted infections (STI). Fergusson et al. (2006) relate the circumcision status and self reported STI history using data from 25-year longitudinal study of a cohort of New Zealand children, known as the Christchurch Health and Development Study.
(a) Out of 356 non-circumcised males, 37 reported having had at least one STI by age 25 . Find the posterior distribution of 1 , the probability a non-circumcised male reports at least one STI by age 25. (Use a beta (1 10) prior for 1.)
(b) Out of the 154 circumcised males, 7 reported having at least one STI by age 25 . Find the posterior distribution of \({ }_{2}\), the probability a circumcised male reports at least one STI by age 25 . (Use a beta (1 10) prior for 2.)
(c) Find the approximate posterior distribution of \(1 \quad{ }_{2}\).
(d) Test the hypothesis
\[
\begin{array}{lllllll}
H_{0}: & 1 & 2 & 0 & \text { versus } & H_{1}: & 1
\end{array}>0
\]
in a Bayesian manner at the \(5 \%\) level of signi cance. What does the result say about the research hypothesis?

Data From Exercise 13

Transcribed Image Text:

Are young New Zealand women having their children at a later age than previous generations? (a) Out of the 314 women in the age group 25 29, 136 report having given birth to their rst child before the age of 25. Find the posterior distribution of 1, the proportion of New Zealand women in that age group who have given birth before age 25. (Use a uniform prior for 1.) (b) Out of the 219 women in the age group 50 54, 135 report having given birth to their rst child before age 25. Find the posterior distribution of 2, the proportion of New Zealand women in that age group who have given birth before age 25. (Use a uniform prior for 2.) (c) Find the approximate posterior distribution of 1 2. (d) Test the hypothesis Ho: 120 versus H: 1 2 <0 in a Bayesian manner at the 1% level of signi cance. Can we conclude that New Zealand women in the generation aged 25 29 have had their rst child at a later age than New Zealand women in the generation aged 50 54?