Testing the Difference Between Proportions 8.4 SECTION 8.4 Testing the Difference Between Proportions 447 What You Should Learn How to perform a two-sample z-test for the difference between two population proportions p1 and p2 Two-Sample z-Test for the Difference Between Proportions Two-Sample z-Test for the Difference Between Proportions In this section, you will learn how to use a z@test to test the difference between two population proportions p1 and p2 using a sample proportion from each population. If a claim is about two population parameters p1 and p2, then some possible pairs of null and alternative hypotheses are eH0: p1 = p2 Ha: p1 ≠ p2 , e H0: p1 … p2 Ha: p1 7 p2 , and e H0: p1 Ú p2 Ha: p1 6 p2 . Regardless of which hypotheses you use, you always assume there is no difference between the population proportions 1p1 = p22. For instance, suppose you want to determine whether the proportion of college students who earn a bachelor of science degree in four years is different from the proportion of college students who earn a bachelor of arts degree in four years. These conditions are necessary to use a z@test to test such a difference. 1. The samples are randomly selected. 2. The samples are independent. 3. The samples are large enough to use a normal sampling distribution. That is, n1p1 Ú 5, n1q1 Ú 5, n2p2 Ú 5, and n2q2 Ú 5. When these conditions are met, the sampling distribution for np 1 − np 2, the difference between the sample proportions, is a normal distribution with mean mnp 1 - np 2 = p1 - p2 and standard error snp 1 - np 2 = Ap1q1 n1 + p2q2 n2 . Notice that you need to know the population proportions to calculate the standard error. Because a hypothesis test for p1 - p2 is based on the assumption that p1 = p2, you can calculate a weighted estimate of p1 and p2 using p = x1 + x2 n1 + n2 where x1 = n1 np 1 and x2 = n2 np 2. With the weighted estimate p, the standard error of the sampling distribution for np 1 - np 2 is snp 1 - np 2 = A pqa 1 n1 + 1 n2b where q = 1 - p. Also, you need to know the population proportions to verify that the samples are large enough to be approximated by the normal distribution. But when determining whether the z@test can be used for the difference between proportions for a binomial experiment, you should use p in place of p1 and p2 and use q in place of q1 and q2. Study Tip The symbols in the table below are used in the z@test for p1 - p2. See Sections 4.2 and 5.5 to review the binomial distribution. Symbol Description p1, p2 Population proportions x1, x2 Number of successes in each sample n1, n2 Size of each sample np 1, np 2 Sample proportions of successes p Weighted estimate of p1 and p2 q Weighted estimate of q1 and q2, q = 1 - p Study Tip You can also write the null and alternative hypotheses as shown below. bH0: p1 - p2 = 0 Ha: p1 - p2 ≠0 bH0: p1 - p2 … 0 Ha: p1 - p2 7 0 bH0: p1 - p2 Ú 0 Ha: p1 - p2 6 0
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