442 CHAPTER 9 Inferences from Two Samples Key Concept In this section we present methods for (1) testing a claim made about two population proportions and (2) constructing a confidence interval estimate of the difference between two population proportions. The methods of this section can also be used with probabilities or the decimal equivalents of percentages. 9-1 Two Proportions Inferences About Two Proportions Objectives 1. Hypothesis Test: Conduct a hypothesis test of a claim about two population proportions. 2. Confidence Interval: Construct a confidence interval estimate of the difference between two population proportions. Notation for Two Proportions For population 1 we let p1 = population proportion p n 1 = x1 n1 (sample proportion) n1 = size of the first sample q n 1 = 1 - p n 1 (complement of p n 1) x1 = number of successes in the first sample The corresponding notations p2, n2, x2, p n 2, and q n 2 apply to population 2. Pooled Sample Proportion The pooled sample proportion is denoted by p and it combines the two sample proportions into one proportion, as shown here: p = x1 + x2 n1 + n2 q = 1 - p Requirements 1. The sample proportions are from two simple random samples. 2. The two samples are independent. (Samples are independent if the sample values selected from one population are not related to or somehow naturally paired or matched with the sample values from the other population.) 3. For each of the two samples, there are at least 5 successes and at least 5 failures. (That is, npn Ú 5 and nqn Ú 5 for each of the two samples). (If the third requirement is not satisfied, alternatives include the resampling methods of bootstrapping or randomization described in Section 9-5, or Fisher’s exact test described in Section 11-2.) Test Statistic for Two Proportions (with H0: p1 = p2) z = 1pn 1 - p n 22 - 1p1 - p22 Apq n1 + pq n2 where p1 - p2 = 0 (assumed in the null hypothesis) Where p = x1 + x2 n1 + n2 (pooled sample proportion) and q = 1 - p KEY ELEMENTS
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