Test statistic for Homogeneity

In order to test for homogeneity we have to look at the difference of each proportion from the extepected value. For example, looking at income and whether the person rents or owns, we can form some hypothses

at < 30,000 income 30% own homes and 70% rent
at > 80,000 income 70% own homes and 30% rent

Thus we can take our sample and find that

at < 30,000 income 40% own homes and 60% rent
at > 80,000 income 60% own homes and 40% rent

To find our test statistic we take the sum of the actual value from the expected, divided by the expected.

sum across i and sum across j (n_ij - e_ij)² / e_ij

We use the resultant test statistic and find where it falls on the x-axis of a chi-square distribution. If it falls so that the area to the right and the area above the point is sufficiently less than 0.05 (or our probability tolerance) we assume the data is homogeneous.

Intro to Tests of Homogeneity

Suppose we wanted to know if the proportions between two populations where similar.

For example:

Are the age ranges between people who rent and people who own homes the same?
Is the income between republican and democratic voters the same?

To answer these questions we have to use tests of homogeneity.
To do this, we take random samples from both groups and record their proportions into categorical variables.