10.3 A dataset on pregnancy rates among girls in 13 north central Florida counties has information on the total in 2005 for each county i on Ti = number of births and yi = number of those for which mother’s age was under 18. Let πi be the probability that a pregnancy in county i is to a mother of age under 18. The logistic-normal model, logit(πi ) = ui + α, has ˆα = −3.24 and ˆσ = 0.33.

a. Find ˆπi for a county estimated to be (i) at the mean, (ii) two standard deviations below the mean, (iii) two standard deviations above the mean on the random effects distribution.

b. For estimating {πi }, what advantage does this model have over the fixed effects model, logit(πi ) = βi?