Suppose that in a particular state consisting of four distinct regions, a random sample of nk voters is obtained from the k th region for . Each voter is then classified according to which candidate (1, 2, or 3) he or she prefers and according to voter registration

. Let denote the proportion of voters in region k who belong in candidate category i and registration category j. The null hypothesis of homogeneous regions is H for all i, j (i.e., the proportion within each candidate/registration combination is the same for all four regions). Assuming that H0 is true, determine and as functions of the observed , and use the general rule of thumb to obtain the number of degrees of freedom for the chi-squared test.