Let X = the number of adult police contacts for a randomly selected individual who previously had at least one such contact prior to age 18. The following frequencies were calculated from information given in the article "Examining the Prevalence of Criminal Desistance" (Criminology, 2003: 423-448); our sample size differs slightly from what was reported because of rounding.
a. Is it plausible that the population distribution of number of contacts is Poisson? Carry out a chisquared test.
b. The cited article did not even entertain the possibility of a Poisson distribution. Instead several other models were proposed. One of these is based on the idea that each individual's number of contacts has a Poisson distribution whose mean value m is itself a random variable having a gamma distribution. This reasoning leads to a generalized negative binomial distribution having two parameters, which must then be estimated from the data. After doing so, the article reported the following estimated probabilities corresponding to the foregoing x values: .6099, .1657, .0838, .0489, .0305, .0197, .0130, .0088, .0060, .0041, .0029, .0020, .0014, .0010, .0007, and .0005. Test the plausibility of this model at significance level .05 by combining all x values exceeding 11 into a single category (this was done in the cited article, which included a P-value).