The following table represents the number of residuals observed in a regression of voter registration rates on income and age over eight districts:
Use the chi-square test to determine whether there is any interaction between location and the tendency of residuals to be positive or negative. If you reject the null hypothesis of no pattern, describe how a researcher might continue the regression analysis.
First, we calculate the expected frequencies of the residuals, given the totals
We use these expected values to calculate the chi-square statistic. For each cell, we divide the squared difference between the observed and expected values by the expected value: e.g. (20 -17.7)2 / 17.7. These values are summed, and their sum comes to 2.885.
We compare this value to the chi-square distribution, for which the critical value is associated with the chosen alpha (0.05) and the degrees of freedom (the number of rows minus one times the number of columns minus one, or 1 * 7 in this case for d.f. 7). 14.067 is larger than our chi-square value of 2.885, so that we fail to reject the null hypothesis of no spatial pattern.
Given the lack of relationship between location and residuals, we need not take any further steps.
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