Table 3.8 lists total attendance (in thousands) and the total number of arrests in a season for soccer teams in the Second Division of the British football league.

a. Let Y denote the number of arrests for a team with total attendance t . Explain why the model E(Y) = μt might be plausible. Show that it has alternative form log[E(Y)/t] = α, where α = log(μ), and express this model with an offset term.

b. Assuming Poisson sampling, fit the model. Report and interpret ˆμ.

c. Plot arrests against attendance, and overlay the prediction equation. Use residuals to identify teams that had a much larger or smaller than expected number of arrests.

d. Now fit the model log[E(Y)/t] = α by assuming a negative binomial distribution. Compare ˆα and its SE to what you got in (a). Based on this information and the estimate of the dispersion parameter and its SE, does the Poisson assumption seem appropriate?