Question

5. a) [6 marks] Suppose a group of 900 smokers (who all wanted to quit smoking) were randomly assigned to receive an antidepressant drug or a placebo for six weeks. Of the 148 patients who received the antidepressant drug, 23 were not smoking one year later. Of the 752 patients who received placebo, 97 were not smoking one year later. Conduct a test of hypotheses (state Hand Ha) to see if taking an antidepressant drug can help smokers to quit smoking. (a = 5%) Here is the data summary Population X n X > = n 1 (drug) 2 (placebo) Total 148 752 900 23 97 b) [2 marks] State your conclusion:

Answer 1

For sample 1, we have that the sample size is N1 = 148, the number of favorable cases is X1 = 23, so then the sample proportion is 1 23 148 = 0.1554 For sample 2, we have that the sample size is N2 = 752, the number of favorable cases is X2 = 97, so then the sample proportion is ôz = x 97 = 0.129 = 752 The value of the pooled proportion is computed as p = Ni+N = 23+97 148+752 = 0.1333 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: pi > P2 This corresponds to a right-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 right-tailed test is ze = 1.64. The rejection region for this right-tailed test is R = {z:z> 1.64} (3) Test Statistics The 2-statistic is computed as follows: 2 = Pi - P2 VP(1-P)(1/nı +1/n2) 0.1554 - 0.129 0.1333 . (1 - 0.1333)(1/148 + 1/752) = 0.864 (4) Decision about the null hypothesis Since it is observed that z = 0.864 < Ze = 1.64, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.1938, and since p = 0.1938 > 0.05, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pı is greater than p2, at the 0.05 significance level. Confidence Interval The 95% confidence interval for p1 - p2 is: -0.037 <P1 - P2<0.09.

As we fail to reject H0 at 5% los, we conclude that the claim is false that the antidepressant drug can help smokers to quit smoking.

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