Question

(Hypothetical) A study is carried out to determine the effect of party affiliation on voting behavior in a certain city. The city is divided up into wards. In each ward, the percentage of registered Democrats who vote must be higher than the percentage of registered Republicans who vote. If true, why? If false, give an example.

Answer 1

The implication is not necessarily true. Take the following example: The city is divided up into two wards. For registered Democrats, there are 100,000 in Ward A and 400,000 in Ward B. There are 500,000 registered Republicans in both wards. In Ward A, 100% (= 100,000) of registered Democrats and 90% (= 450,000) of registered Republicans vote. In Ward B, the figures are 10% (= 40,000) and 9% (= 45,000) respectively. It follows that the percentage of registered Democrats who vote is higher than the percentage of registered Republicans who vote in each ward. On the city level, however, only 28% (= (100,000 + 40,000) / (100,000 + 400,000)) of registered Democrats vote, of which the percentage is less than that of voting registered Republicans ((450,000 + 45,000) / (500,000 + 500,000) = 49.5%). By contradiction, a higher percentage of vote in each ward is not followed by a higher percentage for the city.

Analysis of the failure of implication: The Democratic Party has an unbalanced distribution of members in the wards. Its voting percentage leads significantly in a small population (Ward A) but it only leads a little in a much larger population (Ward B). Since the percentage for the city can be regarded as a sum of the ward percentages weighted by ward population, the effect of the significant lead in Ward A is overcome by the small population. Meanwhile, the Republicans are distributed evenly in each ward. The high (yet lower than that of Democrats) percentage in Ward A receives a fair weight and is thus enough to beat in the city-wide percentage.