Question

The herb ginkgo biloba is commonly used as a treatment to prevent dementia. In a study of its effectiveness, 1554 elderly subjects were given ginkgo and 1596 elderly subjects were given a placebo. Anong those in the ginkgo group, 239 later developed dementia, and among those in the placebo group, 269 developed dementia. Use a 0.05 significance level to test the claim that ginkgo is effective in preventing dementia. Based on your results, is ginkgo effective in preventing dementia?

Answer 1

If we want to test that ginkgo is effective, then we are trying to find out if the proportion of those who had ginkgo and developed dementia is less than the proportion of those who had a placebo.

Let $\hat{p}_1$ = The Proportion of elderly people who developed dementia after having ginkgo = 239 / 1554 = 0.1538

Let $\hat{p}_2$ = The Proportion of elderly people who developed dementia after having a placebo = 269 / 1596 = 0.166

Let $\hat{p}$ = Overall proportion = (239+265)/(1554+1596) = 0.16

1 - $\hat{p}$ = 0.84

$\alpha$ = 0.05

(a) The Hypothesis:

H0: p₁ = p₂

Ha: p₁ < p₂

This is a Left tailed Test.

The Test Statistic:

$Z = \frac{p1-p2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n1}+\frac{1}{n2})}} = \frac{0.1538-0.166}{\sqrt{0.16*0.84*(\frac{1}{1544}+\frac{1}{1596})}} = -0.96$

The p Value:    The p value (Left tail) for Z = -0.96, is; p value = 0.1685

The Critical Value:   The critical value (Left tail) at $\alpha$ = 0.05, Z critical = -1.645

The Decision Rule: If Z observed is <- Z critical Then Reject H0.

Also If the P value is < $\alpha$ , Then Reject H0

The Decision:    Since Z observed (-0.96) is > Z critical (-1.645), We Fail to Reject H0.

Also since P value (0.1685) is > $\alpha$ (0.05), We Fail to Reject H0.

The Conclusion:   There isn't sufficient evidence at the 95% significance level to conclude that ginkgo is effective in preventing dementia in elderly people.