Suppose that in a particular state consisting of four distinctregions, a random sample

Chapter 14, Problem 35

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Suppose that in a particular state consisting of four distinct regions, a random sample of \(n_k\) voters is obtained from the kth 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 \(p_{i j k}\) 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 \(H_0: p_{i j 1}=p_{i j 2}=p_{i j 3}=p_{i j 4}\) for all i, j (i.e., the proportion within each candidate/registration combination is the same for all four regions). Assuming that \(H_0\) is true, determine \(\hat{p}_{i j k}\) and \(\hat{e}_{i j k}\) as functions of the observed \(n_{i j k}\) 's, and use the general rule of thumb to obtain the number of degrees of freedom for the chi-squared test.