Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 102 / 3 / a / Solution

Only four of the twenty provider records are printed, and the full emailed file has not been recovered. The four observed passenger averages have mean $85.75$, median $82.5$, sample standard deviation $17.67$ and range $68$--$110$. Their bus-mile mean is $112.25$, with range $22$--$160$; the highest passenger average belongs to the provider with the lowest bus miles and zero fare. That is a useful diagnostic of differing service patterns, not evidence of a population-wide negative mileage effect.

For the complete dataset, use scatter plots of passenger average against bus miles, fare and population, and inspect population-scaled demographic counts as well as their raw counts. Compare marginal distributions and the sample correlation matrix, especially among population, older residents, low-income households and poverty households: these measure overlapping aspects of area size and composition and can create multicollinearity. Inspect the zero fare directly; a logarithm of fare is undefined there, so do not silently drop it. All four displayed rows belong to the fixed-route group, so they cannot support a between-service comparison.

The figure labels its bus panel as a four-row excerpt. Twenty-provider summaries and exploratory conclusions cannot be determined from these four rows. In particular a full eight-parameter regression has more parameters than observed rows; reporting a fitted full-data model from this excerpt would be unjustified.