Chapter 10.1, Problem 24E

Solution

a)

The significance test and to check if the conditions to carry out this test are met.

Two sample Z test for proportions is used and the conditions are satisfied.

Given:

The number of women got pregnant in treatment group $x_1=34$

The number of women got pregnant in control group $x_2=21$

The total number of women in treatment group $n_1=80$

The total number of women in control group $n_2=80$

P₁ be the proportion of women got pregnant in treatment group

P₂ be the proportion of women got pregnant in control group

The null and alternate hypothesis are

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

Concept used:The conditions to satisfy to run two sample Z test for difference in proportions are

1) The sample should be random

2) np = 10

3) n(1-p) = 10

4) The observations in each group should be independent.

Calculation:$\begin{array}{l}{p}_1=\frac{34}{80}=0.43\\ p_2=\frac{21}{80}=0.26\end{array}$

  $\begin{array}{l}{n}_1{p}_1=80*0.43=34\\ n_1{p}_1>10\\ n_1(1-p_1)=80*(1-0.34)=46\\ n_1(1-p_1)>10\\ n_2{p}_2=80*0.26=21\\ n_2{p}_2>10\\ n_2(1-p_2)=80*(1-0.26)=59\\ n_2(1-p_2)>10\end{array}$

The experiment is randomized.

As the experiment is randomized, the groups of women are treated as independent. If one woman got pregnant it gives no information about another woman. So individual observations in each group are also considered to be independent.

All conditions are met so we can use two sample Z test for difference in proportions.

b)

Interpret the p value of the significance test..

Given:The p value of the significance test is 0.0152

Explanation:The p value is probability that null hypothesis is true. The p value of 0.0152 means that there is 1.52% chance that there is no difference in pregnancy rates of women from treatment group (receiving acupuncture) and control group.

c)

The conclusion drawn in significance test at 5% level of significance..

Given:The p value of the significance test is 0.0152 and level of significance is 0.05

Explanation:The p value 0.0152 is less than the level of significance 0.05. There is sufficient evidence at 5% level of significance that the null hypothesis is rejected. We conclude that the proportion of pregnancies among women from treatment group are higher than the control group. So proportion of pregnancies among women who receive acupuncture is higher than who do not receive acupuncture.

d)

The flaw in the experiment which prevent us drawing a cause effect conclusion.

Explanation:As the women can see acupuncture being done to them, the women in control group knew that they had not received acupuncture. This may affect the behavior in such a way as to affect whether they became pregnant or not. This is the flaw in this experiment.