Question

The Acme Drug Company develops a new drug, designed to prevent colds. The company states that the drug is more effective for women than for men. To test this claim, they choose a simple random sample of 100 women and 200 men from a population of 100,000 volunteers.

At the end of the study, 38 of the women caught a cold; and 102 of the men caught a cold. Based on these findings, can we conclude that the drug is more effective for women than for men? Use a 0.05 level of significance.

a) State the null hypothesis and the alternative hypothesis.

            b) State the test that should be used to test this hypothesis?

            c) Compute the appropriate test statistic for this test.

            d) Determine the critical value(s) for the significance level.

            e) Determine the p-value for this test.

            f) Based on the p-value or critical value(s) you found above, what is your conclusion. Explain.

            g) Find a 95% confidence interval and explain how this supports your conclusion.

Answer 1

For sample 1, we have that the sample size is N1 = 100, the number of favorable cases is X1 = 38, so then the sample proportion is p1 = 100 = 0.38 For sample 2, we have that the sample size is N2 = 200, the number of favorable cases is X2 = 102, so then the sample proportion is p2 = 102 200 = 0.51 The value of the pooled proportion is computed as p = * =* = 38+102 100+200 = 0.4667 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: P1 P2 This corresponds to a left-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 left-tailed testis ze = -1.64. The rejection region for this left-tailed test is R = {:::< -1.64} (3) Test Statistics The Z-statistic is computed as follows: P1 - 2 Vp(1-7)(1/ni + 1/n2) 0.38 – 0.51 70.4667. (1 - 0.4667)(1/100 +1/200) -2.128 (4) Decision about the null hypothesis 1.64, it is then concluded Since it is observed that := -2.128 <%= that the null hypothesis is rejected. Using the P-value approach: The p-value is p = 0.0 since p = 0.0167 <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 puis less than p2, at the 0.05 significance level. Confidence Interval The 95% confidence interval for pi - p2 is: -0.248 <pi - p2 -0.012.

As null hypothesis is Rejected at 5% los, we conclude that the claim is true that the drug is more effective on women than men.

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