Illuminating Statistical Analysis Using Scenarios and Simulations

31Two-Way Proportions: Homogeneity

A survey was conducted in the Flowing Wells (FW) community in which 100 random members recorded their political affiliation. The same survey was conducted in the Artesian Wells (AW) community asking 100 random members their political affiliation. Below are the observed counts of Democrats, Republicans, and Independents for FW and AW.

FW: 40, 30, and 30.

AW: 25, 30, and 45.

We want to assess whether the two communities might have different population proportions of Democrats, Republicans, and Independents.

Null hypothesis: No difference between the FW and AW communities' population proportions.

Rephrasing: How likely is it to get these sample counts if in fact FW and the AW have the same population proportions?

This scenario is an extension of the scenario in Chapter 12 in which we wanted to know if there was a proportion difference between FW and AW for a binomial variable. Now we are concerned with a multinomial variable.

To analyze this scenario, we can use the Chi-squared (X²) Test for Homogeneity. A Chi-squared calculator tells us that the Chi-squared statistic is 6.462, degrees of freedom is two, and the -value is 0.0395. Reject the null hypothesis that the two communities have the same population proportions of Democrats, Republicans, and Independents.

Detailed below is the arithmetic that yields the Chi-square of 6.462.

Table 31.1 ...