448 CHAPTER 8 Hypothesis Testing with Two Samples When the sampling distribution for np 1 - np 2 is normal, you can use a two-sample z@test to test the difference between two population proportions p1 and p2. A two-sample z@test is used to test the difference between two population proportions p1 and p2 when these conditions are met. 1. The samples are random. 2. The samples are independent. 3. The quantities n1p, n1q, n2p, and n2q are at least 5. The test statistic is np 1 - np 2. The standardized test statistic is z = 1 np 1 - np 22 - 1p1 - p22 B pqa 1 n1 + 1 n2b where p = x1 + x2 n1 + n2 and q = 1 - p. Two-Sample z-Test for the Difference Between Proportions If the null hypothesis states p1 = p2, p1 … p2, or p1 Ú p2, then p1 = p2 is assumed and the expression p1 - p2 is equal to 0 in the preceding test. Using a Two-Sample z-Test for the Difference Between Proportions In Words In Symbols 1. Verify that the samples are random and independent. 2. Find the weighted estimate of p1 p = x1 + x2 n1 + n2 , q = 1 - p and p2. Verify that n1p, n1q, n2p, and n2q are at least 5. 3. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 4. Specify the level of significance. Identify a. 5. Determine the critical value(s). Use Table 4 in Appendix B. 6. Determine the rejection region(s). 7. Find the standardized test statistic z = 1 np 1 - np 22 - 1p1 - p22 B pqa 1 n1 + 1 n2b and sketch the sampling distribution. 8. Make a decision to reject or fail to If z is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES A hypothesis test for the difference between proportions can also be performed using P@values. Use the guidelines above, skipping Steps 5 and 6. After finding the standardized test statistic, use Table 4 in Appendix B to calculate the P@value. Then make a decision to reject or fail to reject the null hypothesis. If P is less than or equal to a, then reject H0. Otherwise, fail to reject H0. Picturing the World A medical research team conducted a study to test whether the administration of vitamin D reduces the incidence of acute kidney injury (AKI) after cardiac surgery. In the study, 50 patients received vitamin D and 61 patients received a placebo. The results are shown below. (Source: Cardiorenal Medicine) Vitamin D Placebo Sustained Acute Kidney Injury 12% 16.4% At A = 0.05, can you support the claim that the administration of vitamin D reduces the incidence of AKI after cardiac surgery? Study Tip To simplify the calculation of z, you can round the values of p, q, p1 n , and p 2 n to four decimal places, as shown in Examples 1 and 2.
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