Chapter 10.1, Problem 23E

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 $x_1=44$

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

The total number of women in treatment group $n_1=88$

The total number of women in control group $n_2=81$

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:$\begin{array}{l}{p}_1=\frac{44}{88}=0.5\\ p_2=\frac{21}{81}=0.26\end{array}$

  $\begin{array}{l}{n}_1{p}_1=88*0.5=44\\ n_1{p}_1>10\\ n_1(1-p_1)=88*(1-0.5)=44\\ n_1(1-p_1)>10\\ n_2{p}_2=81*0.26=21\\ n_2{p}_2>10\\ n_2(1-p_2)=81*(1-0.26)=60\\ n_2(1-p_2)>10\end{array}$

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 0.0007

Explanation:The p value is probability that null hypothesis is true. The p value of 0.0007 means that there is 0.07% chance that there is no difference in pregnancy rates of women from treatment group and control group.

c)

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

Given:The p value of the significance test is 0.0007 and level of significance is 0.05

Explanation:The p value 0.0007 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 prayed for are higher than who does not prayed for.

d)

The importance of women in the study not knowing if they were being prayed for.

Explanation:If the women had known that they were being prayed for then this might affected their behavior knowingly or unconsciously, that would have affected whether they became pregnant or not.