a. One program is designed for divorce counseling: the key feature of the program is its counselors, who are married couples working in teams. About half of all clients have been randomly assigned to this special program and half to the regular program, and the proportion of cases that eventually ended in divorce was recorded for both. The results for random samples of couples from both programs are reported here In terms of preventing divorce, did the new program work?
Sample 1 (Special Program) 
Sample 2 (Regular Program) 




b. The agency is also experimenting with a program that includes peer counseling for depressed teenagers. About half of all clients were randomly assigned to the new program. After a year, a random sample of teens from the new program was compared with a random sample that received standard counseling. In terms of the percentage of children who were judged to be “much imp roved,” did the new program work?
Sample 1 (Peer Counseling) 
Sample 2 (Standard Program) 




(a)
To find:
The significant difference between two sample proportions.
Given:
The sample statistics is given in the table below,
Sample 1 (Special Program) 
Sample 2 (Regular Program) 
Approach:
The five step model for hypothesis testing is,
Step 1. Making assumptions and meeting test requirements.
Step 2. Stating the null hypothesis.
Step 3. Selecting the sampling distribution and establishing the critical region.
Step 4. Computing test statistics.
Step 5. Making a decision and interpreting the results of the test.
Formula used:
The formula to calculate the sampling distribution of the differences in sample proportions of large samples is given by,
Where,
Where,
And
Calculation:
As the significant difference in the sample proportions is to be determined and a direction is predicted, a one tailed test is applied.
Follow the steps for twosample testing as,
Step 1. Making assumptions and meeting test requirements.
Model:
Independent random samples.
Level of measurement is nominal.
Sampling distribution is Normal.
Step 2. Stating the null hypothesis.
The statement of the null hypothesis is that there is no significant difference in the samples of the population. Thus, the null and the alternative hypotheses are,
Step 3. Selecting the sampling distribution and establishing the critical region.
Since, the sample size is large, Z distribution can be used.
Thus, the sampling distribution is Z distribution.
The level of significance is,
Area of critical region is,
Step 4. Computing test statistics.
The population standard deviations are unknown.
The formula to calculate
Substitute 0.53 for
(b)
To find:
The significant difference between two sample proportions.
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