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 k 5 1, 2, 3, 4. Each voter is then classified according to which candidate (1, 2, or 3) he or she prefers and according to voter registration (1 = Dem., 2 = Rep., 3 5 Indep.). Let pijk 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 H0: pij1 = pij2 = pij3 = pij4 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 ijk and ê ijk as functions of the observed nijk's, and use the general rule of thumb to obtain the number of degrees of freedom for the chi-squared test.