Statistical Inference: A Short Course

13.6 Testing k Proportions

We now turn to a special case of Equation (13.14) in that we examine the instance in which we want to test for the significance of the difference among k population proportions p_i, i = 1, . . ., k. In this regard, suppose we have k independent random samples and that X₁, . . ., X_k comprise a set of independent binomial random variables with the parameters p₁ and n₁; p₂ and n₂; . . .; and p_k and n_k, respectively, where p_i, i = 1, . . ., k, is the proportion of successes in the ith population. Here X_i depicts the number of successes obtained in a sample of size n_i, i = 1, . . ., k.

Let us arrange the observed number of successes and failures for the k independent random samples in the following k × 2 table (Table 13.8). Here the 2k entries within this table are the observed cell frequencies o_ij, i = 1, . . ., k; j = 1,2. Our objective is to test H₀: p₁ = p₂ = . . . = p_k = p_o, against H₁: p_i ≠ p_o for at least one i = 1, . . ., k, where p_o is the null value of p_i. Under H₀, the expected number of successes for sample i is n_ip_o, i = 1, . . . k (since p_o = X_i/n_i); and the expected number of failures for sample i is n_i(1 − p_o), i = 1, . . .,k (since ). In this regard, the expected cell frequencies for columns 1 and 2 are, respectively, and . So given p_o, Equation ...