Victims of homicide

A recent General Social Survey asked subjects, “Within the past 12 months, how many people have you known personally that were victims of homicide”? The sample mean for the 159 blacks was 0.522, with a variance of 1.150. The sample mean for the 1149 whites was 0.092, with a variance of 0.155.
a. Let yij denote the response for subject j of race i, and let μij E(Yij ). The Poisson model log(μij ) α+βxij with x1j 1 (blacks) and x2j
0 (whites) has fit log(μij )= 2.38 + 1.733xij . Show that the estimated population means are 0.522 for blacks and 0.092 for whites, which are the sample means.
b. For the Poisson GLM, the standard error of βˆ is 0.147. Show that the Wald 95% confidence interval for the ratio of means for blacks and whites is (4.2, 7.5). [Hint: Note that β is the log of the ratio of the means.]
c. The negative binomial loglinear model has the same estimates as in (a), but the standard error of βˆ increases to 0.238 and the Wald 95%
confidence interval for the ratio of means is (3.5, 9.0). Based on the sam- ple means and variances, which confidence interval is more believable? Why?
d. The negative binomial model has Dˆ 4.94 (SE =1.00). Explain why this shows strong evidence that the negative binomial GLM is more appropriate than the Poisson GLM.

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