Question

Hormone Therapy and Dementia. An issue of Science News (Vol. 163, No. 22, pp. 341-342) 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 greater risk for dementia than those who do not?

Short answer

There is $90\%$confident that the difference in dementia risk rates for healthy women over $65$years old who take hormone-replacement therapy and those who do not lies between$0.00317$and $0.01448$

Step-by-step solution

Given information

Null hypothesis: $H_0:p_1=p_2$

the to conclude that healthy women over 65
do not.

Alternative hypothesis:

$H_a:p_1>p_2$

That is, 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 (a) Step 2: Explanation

Find the value of test statistic and p-value by using MINITAB. MINITAB procedure:

MINITAB procedure

Step 1: Choose Stat > Basic Statistics > 2 Proportions.

Step 2: Choose Summarized data.
Step 3: In First sample, enter Trials as $2,229$and Events as$40$.
Step 4: In Second sample, enter Trials as $2,303$and Events as21.

Step 5: Check Perform hypothesis test. In Hypothesized proportion, enter 0 .
Step 6: Choose Options
Step 7: In Confidence level, enter$95$
Step 8: In Alternative, select greater than.
Step 9: Click OK.

From the MINITAB output, the test statistic is $2.58$and the p-value is $0.005$

Part (a) Step 3: Explanation

Use the significance level $\alpha =0.05$

Here, $p$-value is lesser than the level of significance

That is, p-value $(=0.005)<\alpha (=0.05)$

hypothesis $\left(H_0\right)$at $\alpha =0.05$
Hence, The data provide enough evidence to conclude that healthy women over the age of 65 who use hormone replacement therapy are more likely to develop dementia than those who do not.

Part (a) Step 4: Explanation

Find the $90\%$confidence interval for the difference in dementia risk rates for healthy women over

Step 1: Choose Stat > Basic Statistics > 2 Proportions.
Step 2: Choose Summarized data.
Step 3: In First sample, enter Trials as $2,229$and Events as $40$
Step 4: In Second sample, enter Trials as $2,303$and Events as $21$
Step 5: Check Perform hypothesis test. In Hypothesized proportion, enter 0 .
Step 7: In Confidence level, enter $90$
Step 8: In Alternative, select not equal.
Step 9: Click OK.

Part (c) Step 5: Explanation

MINITAB output

Test and Cl for Two Proportions

From the MINITAB output, the $90\%$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 is $(0.00317,0.01448).$

There is $90\%$confident that the difference in dementia risk rates for healthy women over 65 years old who take hormone-replacement therapy and those who do not lies between$0.00317$and $0.01448$.