Chapter 10.1, Problem 28E

(a)

To determine

To name the appropriate test and check that the conditions for carrying out this test are met.

Answer to Problem 28E

The two sample z test is the appropriate test.

Explanation of Solution

Given:

It is given that a study of whether ancient Chinese art of acupuncture could help infertile women become pregnant or not is conducted. And it is also given the hypothesis that:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Therefore, according to the study we want to execute a test about comparing two populations mean this can be done by using the two sample z test for the difference in two proportions.

Conditions to be satisfied:

There are three conditions to be satisfied:

Random: It is satisfied because the women are selected randomly for the experiment.

Independent: It can be assumed because the women were randomly assigned to a group.

Normal: It is satisfied because there are $34$ successes and $80-34=46$ failures in the first sample and there are $21$ successes and $80-21=59$ failures in the second sample, which are all at least ten.

Thus, all the conditions are satisfied.

(b)

To determine

To interpret the P -value for this context.

Answer to Problem 28E

The probability of obtaining a difference between the proportions as in this sample or more extreme is $1.52\%$ , if the population proportions are equal.

Explanation of Solution

It is given that the appropriate test given in part (a) yields P value as $0.0152$ . Thus, we have,

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\\ P=0.0152=1.52\%\end{array}$

Now, the P value is the probability of obtaining the value of the test statistics or a value more extreme if the null hypothesis is true.

Thus, we interpret from the P value that it is the probability of obtaining a difference between the proportions as in this sample or more extreme is $1.52\%$ , if the population proportions are equal.

(c)

To determine

To find out what conclusion the researcher draw at the $\alpha =0.05$ significance level.

Answer to Problem 28E

There is sufficient evidence that the pregnancy rate is higher for women who received acupuncture.

Explanation of Solution

It is given that the appropriate test given in part (a) yields P value as $0.0152$ . Thus, we have,

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\\ P=0.0152=1.52\%\end{array}$

And as we know that if the P value is smaller than the significance level then we reject the null hypothesis. Thus, we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is sufficient evidence that the pregnancy rate is higher for women who received acupuncture.

(d)

To determine

To explain why this is important.

Explanation of Solution

In this study it is mentioned that the women knew that whether or not they received acupuncture. This is important for a study to hide this because whether the women know or not does not have an effect on the results of the experiment since they are unable to control their own fertility. Also, the women would need to know anyway because they would feel the acupuncture as opposed to lying still and not feeling anything.