Question

1. The herb ginkgo biloba is commonly used as a treatment to prevent" dementia. In a study of the effectiveness of this treatment, 1545 elderly subjects were given ginkgo and 1524 elderly subjects were given a placebo. Among those in the ginkgo treatment group, 246 later developed dementia, and among those in the placebo group, 277 later developed dementia. We want to use a 0.01 significance level to test the claim that ginkgo is effective in preventing dementia. a. Find ni, Xi, p-hat, and q-hat, for the first sample. Also, find n2, X2, p-hat2 and q-hat2 for the second sample.

Answer 1

Sample 1 : gingko treatment group

Sample 2 : Placebo treatment group

p1 : proportion of elderly subject developed dementia in gingko treatment group

p2 : proportion of elderly subject developed dementia in Placebo treatment group

For sample 1, we have that the sample size is N1 = 1545, the number of favorable cases is X1 = 246, so then the sample proportion is ội = Xi = 23:45 = 0.1592 For sample 2, we have that the sample size is N2 = 1524, the number of favorable 277 cases is X2 = 277, so then the sample proportion is p2 = 1524 = 0.1818 The value of the pooled proportion is computed as p = x1+ 246+277 0.1704 1545+1524 Also, the given significance level is a = 0.01. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho:pı > P2 Ha: Pi<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 = 0.01, and the critical value for a left-tailed test is ze = -2.33. The rejection region for this left-tailed test is R = {z:z < -2.33} (3) Test Statistics The Z-statistic is computed as follows: Pi - P2 (1-P)(1/nı + 1/n2) 0.1592 -0.1818 = -1.66 0.1704 . (1 – 0.1704)(1/1545 +1/1524) (4) Decision about the null hypothesis Since it is observed that z = -1.66 > ze = -2.33, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.0484, and since p = 0.0484 > 0.01, it is concluded that the null hypothesis is not rejected.

We Failed to reject Ho, we conclude that there is no enough evidence to claim that the gingko if effective in preventing dementia.

Z-test for Two Proportions, z = -1.66, p = 0.0484 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

PL??