9-1 Two Proportions 445 Step 6: Because we assume in the null hypothesis that p1 = p2, the value of p1 - p2 is 0 in the following calculation of the test statistic: z = 1pn 1 - p n 22 - 1p1 - p22 Ap q n1 + p q n2 = a 79 438 - 44 446b - 0 B10.13914027210.860859732 438 + 10.13914027210.860859732 446 = 3.51 This is a two-tailed test, so the P-value is twice the area to the right of the test statistic z = 3.51 as shown in Figure 8-3 on page 380. Refer to Table A-2 and find that the area to the right of z = 3.51 is 0.0001, so the P-value is 210.00012 = 0.0002. Technology provides a more accurate P-value of 0.00045 (as shown in the preceding Statdisk display). The test statistic and P-value are shown in Figure 9-1(a). FIGURE 9-1 Hypothesis Test with Two Proportions Test Statistic: z 5 3.51 p1 2 p2 5 0 or z 5 0 a/2 5 0.025 a/2 5 0.025 z 5 1.96 z 5 21.96 (b) Critical Value Method Test Statistic: z 5 3.51 p1 2 p2 5 0 or z 5 0 P-value: For a two-tailed test, double this area. (a) P-Value Method Step 7: Because the P-value of 0.00045 is less than the significance level of a = 0.05, we reject the null hypothesis of p1 = p2. (“If the P is low, the null must go.”) INTERPRETATION We must address the original claim that there is no difference in success rates between the two treatment groups. Because we reject the null hypothesis, we conclude that there is sufficient evidence to warrant rejection of the claim that p1 = p2. It appears that there is a significant difference between success rates in the two treatment groups. But wait, we’re not done. Based on this hypothesis test, it appears that the e-cigarette treatment is better than the nicotine replacement treatment. We can verify that with a right-tailed test of p1 7 p2, the test statistic is the same z = 3.51 and the P-value will be 0.00022 (the P-value is 0.00045 for the two-tailed test, and it is 0.00022 for the right-tailed test). However, even the e-cigarette treatment has a success rate of only 18.0%, indicating that smokers using this treatment have less than a 20% chance of success in their attempt to stop smoking. The e-cigarette treatment might be better, but it is not that effective. YOUR TURN. Do Part (a) of Exercise 7 “Buttered Toast Drop Test.”
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