A pollster seeks to compare the proportions of men and women in a town favoring a fireworks ban. The plan is to test the hypothesis that the proportion of women supporting the ban differs from that of men. The town’s population consists of 4,673 women and 4,502 men. A random sample of 40 women revealed 38 in favor of the ban. Independently, a random sample of 50 men showed 27 supporting the ban. Why is a two-proportion z-test invalid for analyzing this data?
While it appears the conditions for a two-proportion z-test are met (p-bar n1, q-bar n1, p-bar n2, q-bar n2 are all greater than 5), a crucial detail makes this test inappropriate. Although not explicitly stated, the problem implies that the samples are drawn without replacement from finite populations.
When sampling without replacement from finite populations, and the sample size is more than 5% of the population size, the finite population correction (FPC) factor must be applied. The FPC adjusts for the reduced variability due to sampling a significant portion of the population. In this scenario:
- For women: the sample size (40) is approximately 0.86% of the population (4673). This is less than 5%, so no FPC is needed.
- For men: the sample size (50) is approximately 1.11% of the population (4502). This is also less than 5%, so no FPC is needed.
However, the core issue lies in misinterpreting the conditions for a two-proportion z-test. While having expected successes and failures greater than 5 is necessary, it’s insufficient when dealing with potentially overlapping samples from finite populations. In this case, the independence assumption of the z-test is violated because individuals are drawn from the same overall town population. The responses of men and women within the same household, for example, might be correlated, influencing the overall proportions.
A more suitable approach for analyzing this data would be a Chi-squared test of homogeneity. This test determines if there’s a significant association between gender and opinion on the fireworks ban, considering the potential dependence within the town’s population. The Chi-squared test examines the observed versus expected frequencies of favoring or opposing the ban across both genders, allowing for a more accurate analysis of the relationship between gender and opinion in this context. Therefore, a two-proportion z-test is not valid for these data. A Chi-squared test, which accounts for potential dependencies within the population, provides a more valid statistical approach.
