Suppose that in a particular state consisting of four distinct regions, a random sample...

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 = 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 = Indep.) . Let p_ijkdenote the proportion of voters in region k who belong in candidate category i and registration category j. The null hypothesis of homogeneous regions is as functions of the observed n_ijk’s, and use the general rule of thumb to obtain the number of degrees of freedom for the chi-squared test.