Question

1. A study published in the New England Journal of Medicine in 1988 provided the prevalence of myocardial infarctions in individuals taking aspirin vs. placebo. They found that of the individuals receiving placebo 189 out of 10,845 experienced MI. Of the 10,933 receiving aspirin 104 experienced MI. a) Construct a 95% confidence interval for the proportion of individuals receiving placebo that experience MI. b) Test whether the proportion of individuals experiencing Mi is the same for individuals taking aspirin vs. placebo, use a = .05.

Answer 1

1.

a.

Favorable Cases X = 189 Sample Size N = 10845 The sample proportion is computed as follows, based on the sample size N = 10845 and the number of favorable cases X = 189: X N 189 10845 = 0.017 = 0.05 is zc = 21-a/2 = 1.96. The corresponding confidence interval is computed The critical value for a as shown below: (1-P) p(1-P) CI(Proportion) - Zc + zc n n 0.017 1.96 x 0.017(1 -0.017) ,0.017 + 1.96 x 10845 0.017(1 – 0.017) 10845 (0.015, 0.02)

b.

For sample 1, we have that the sample size is N1 = 10845, the number of favorable cases is X1 = 189, so then the sample proportion is ĝi = x; = 10895 = 0.0174 For sample 2, we have that the sample size is N2 10933, the number of favorable cases is X2 so then the sample proportion is P2 104 NE = 0.0095 10933 104, = The value of the pooled proportion is computed as p = X1+X = 189+104 10845+10933 0.0135 Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = P2 Ha:py = P2 This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a two- tailed test is ze = 1.96. The rejection region for this two-tailed test is R = {z : [2] > 1.96} (3) Test Statistics The Z-statistic is computed as follows: Z P1 - 2 Vp(1 - b)(1/11+1/09) = 0.0174 - 0.0095 0.0135. (1 - 0.0135)(1/10845 + 1/10933) = 5.069

(4) Decision about the null hypothesis Since it is observed that |z| = 5.069 > ze = 1.96, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p= 0, and since p=0 < 0.05, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1 is different than p2, at the 0.05 significance level.

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