Statistics

Ian Stewart wrote:The percentage of households with an annual income of more than $40,000 is higher in Merton county than in any other county. However, the percentage of households with an annual income of $60,000 or more is highest in Sommer county. If the statements above are true, which of the following can properly be concluded on the basis of them?

B. Some households in Merton county have an annual income between $40,000 and $60,000.

Does this need to be true? Let's make it false: No household in Merton has an income between 40k and 60k, and see what happens:

If x% of households in Merton earn more than $40k, then x% would need to earn more than $60k - not a single Merton household earns between $40k and $60k. Now, suppose y% of Sommer county households make over $40k, and z% make over $60k. Clearly y >= z; fewer people earn over $60k than earn over $40k. But we know from the question that x > y (a higher percentage in Merton earn over $40k than in any other county). If x > y, and y >= z, then x > z, which contradicts information in the question (that Sommer has a higher percentage of $60k+ earners).

That is, if we assume B is false, we get a contradiction, so B must be true.