Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 1

The examination used separately emailed files. For this question a public Guerry data archive and its CSV mirror permit a documented reconstruction: retain the 85 records with a recorded region and select the specified columns. All six printed records match exactly, and each of the five regions has 17 departments. The numerical results below refer to that reconstruction; the remaining rows cannot be checked against the unavailable emailed file.

Treat department identifiers as labels and region as a categorical variable. The following summaries show the scale and spread of the quantitative variables:

A larger population-per-crime value means a lower crime rate, and a smaller wealth rank means greater recorded wealth. Donations are markedly right-skewed, as the large gap between mean and median and the upper tail indicate; rank is an ordinal measure rather than a monetary difference. For the ratio analysed next, the median is $2.683$, the interquartile interval is $[1.829,3.515]$, and the range is $[0.643,10.194]$.

Regional median ratios are $2.763$ in C, $2.502$ in E, $3.979$ in N, $1.373$ in S, and $2.937$ in W. The regional box plots and wealth-versus-log-ratio scatter plot in the figure expose this geographical contrast and an inverse association with wealth rank. North has the highest typical property-to-person crime ratio and South the lowest in the reconstructed data. These summaries are descriptive departmental comparisons, not causal or individual-level conclusions.

If a department has population $P$, $C_{mpers}$ crimes against persons and $C_{mprop}$ crimes against property, cancellation of the common numerator givesThus $y$ is a property-to-person crime count ratio, not its reciprocal. Both counts must be positive for the printed ratio to be defined.

For the reconstructed 85-row data, fit a normal linear model for the Box–Cox transformation $T_{\lambda }(y)$, including four region contrasts plus Literacy, Donations and Wealth. With $z_i(\lambda )=(y_i^{\lambda }-1)/\lambda$ and $z_i(0)=\ln y_i$, the profile likelihood, up to constants, isThe second term is the transformation's Jacobian determinant; omitting it would incorrectly compare different response scales. Maximization gives $\hat{\lambda }=0.143$, with approximate 95% likelihood-ratio confidence interval $[-0.120,0.406]$. The logarithmic transformation is inside this interval, whereas the raw-scale choice $\lambda =1$ is far outside: the twice-log-likelihood drops at zero and one are $1.14$ and $39.54$, respectively. A log-response model is therefore a simple supported choice.

In the full log model, partial F-tests give $F_{4,77}=13.15$ for Region ($p=3.3\times 10^{-8}$), $F_{1,77}=27.99$ for Wealth ($p=1.1\times 10^{-6}$), and $p=0.786,0.682$ for Literacy and Donations. Dropping Literacy and Donations together gives $F_{2,77}=0.109$ and $p=0.897$. A parsimonious final normal linear model, with C as reference and $W=\mathrm{Wealth}/10$, isThe standard errors in that order are $0.1056,0.1105,0.1158,0.1102,0.1104,0.01539$; residual standard deviation is $0.3212$ on 79 degrees of freedom and $R^2=0.626$. Use residual-versus-fitted and quantile-quantile plots, influence diagnostics and checks for remaining curvature to assess the working Gaussian model; the small p-values alone do not validate its assumptions.

Holding Region fixed, worsening wealth rank by ten positions multiplies the fitted conditional median ratio by $e^{-0.08539}=0.918$. Relative to C at the same wealth rank, the fitted median multipliers are $1.081$ for E, $1.375$ for N, $0.563$ for S and $1.212$ for W. Region and wealth rank explain much of the variation; Literacy and Donations add little once those terms are included. These are associations in the reconstructed dataset. Back-transforming the fitted log mean gives a conditional median under lognormal errors; estimating an original-scale mean requires bias correction after an inverse transformation. Also, the public archive's metadata describe some historical measures differently from the exam wording, so the fitted Donations effect is interpreted only on its recorded scale.