= Solution
If a department has population $P$, $C_{
m pers}$ crimes against persons and $C_{
m prop}$ crimes against property, cancellation of the common numerator gives
$$
\boxed{y=\frac{P/C_{\rm pers}}{P/C_{\rm prop}}=\frac{C_{\rm prop}}{C_{\rm pers}}.}
$$
Thus $y$ 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, is
$$
\ell_p(\lambda)=-\frac{85}{2}\ln\left[\frac{\operatorname{RSS}(\lambda)}{85}\right]
+(\lambda-1)\sum_i\ln y_i.
$$
The second term is the transformation's <Jacobian determinant>; omitting it would incorrectly compare different response scales. Maximization gives $\widehat\lambda=0.143$, 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 ($p=3.3\times10^{-8}$), $F_{1,77}=27.99$ for Wealth ($p=1.1\times10^{-6}$), 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 $W=\mathrm{Wealth}/10$, is
$$
\boxed{\widehat{\ln y}=1.2956+0.07794\mathbf1_E+0.31846\mathbf1_N
-0.57482\mathbf1_S+0.19198\mathbf1_W-0.08539W.}
$$
The 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. \b[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.
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