Question

An issue of Science News reported that the Women's Health Initiative cast doubts on the benefit of hormone-replacement therapy. Researchers randomly divided \(4532\) healthy women over the age of \(65\) years into two groups. One group, consisting of \(2229\) women received hormone-replacement therapy; the other group consisting of \(2303\) women, received placebo. Over \(5\) years, \(40\) of the women receiving the hormone-replacement therapy were diagnosed with dementia, compared with \(21\) of those getting placebo.

a. At the \(5%\) significance level, do the data provide sufficient evidence to conclude that healthy women over \(65\) years old who take hormone-replacement therapy are at great risk for dementia than those who do not?

b. Determine and interpret a \(90%\) confidence interval for the difference in dementia risk rates for healthy women over \(65\) years old who take hormone-replacement therapy and those who do not.

Short answer

Part a. The data provide sufficient evidence to conclude that healthy women over years old who take hormone replacement therapy are at greater risk for dementia than those who do not.

Part b. There is \(90%\) interval for the difference between the proportions is \((0.00317, 0.01448)\).

Step-by-step solution

Part a. Step 1. Given information

The given values are, \(x_{1}=40, n_{1}=2229, x_{2}=21, n_{2}=2303, \alpha=0.05\)

Part a. Step 2. Calculation

Using MINITAB output.

MINITAB output: Test and CI two proportions

From the MINITAB, the test statistic is \(2.58\) and the \(p-\)value is \(0.005\).

\(p-\)value is lesser than the level of significance.

\(p-\)value \((=0.005)<a(=0.05)\)

Using the rejection rule, it can be concluded that there is evidence to reject null hypothesis \(H_{0}\): at \(\alpha =0.005\).

Therefore, the data provide sufficient evidence to conclude that healthy women over \(65\) years old who take hormone replacement therapy are at greater risk for dementia than those who do not.

Part b. Step 1. Explanation

Using the INITAB output.

MINITAB output: Test and CI for two proportions.

From the MINITAB output, the \(90%\) confidence interval for the difference in dementia risk rate for healthy women over \(65\) years old who take hormone replacement therapy and those who do not is \((0.00317, 0.01448)\).

Therefore, there is \(90%\) interval for the difference between the proportions is \((0.00317, 0.01448)\).