Illuminating Statistical Analysis Using Scenarios and Simulations

28Multiple Proportions and the X²-Distribution

Statistical Scenario—Political Affiliation

You are wondering about political affiliations in your community, so you randomly survey 90 community members and ask each one if they are a Democrat, a Republican, or an Independent.

Your null hypothesis is that your community's political affiliations are perfectly balanced (evenly split), with one-third in each category.

Under the null hypothesis, your expected sample counts are 30, 30, and 30.

Say that the actual sample counts you observe turn out to be 40, 30, and 20.

Should you reject your null hypothesis?

How likely is it to randomly sample 40, 30, 20 from an evenly split population?

Let's expand from scenarios involving binomial variables, as we investigated in Part I, to scenarios involving multinomial variables. Political affiliation is a multinomial variable with three possible values: D, R, and I. How would we go about assessing whether a community is evenly split between the three, with 33.3% in each category? You have a random sample of 90 community members with 40 D, 30 R, and 20 I. If the null hypothesis is true, we expect about 30, 30, and 30 sample members in each of the three political affiliation categories. The word “about” needs to be more exact: would 35, 30, 25 be close enough? 37, 30, 23? 40, 30, 20?

We need to (re)invent a sample statistic that reflects how far the actual observed counts are from our expected counts. If we sum up the differences between the ...