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S W A large counseling center needs to evaluate several experimental programs. Write a paragraph summarizing the results of these tests. Did the new programs work? 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) P s 1 = 0.53 P s 2 = 0.59 N 1 = 78 N 2 = 82 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) P s 1 = 0.10 P s 2 = 0.15 N 1 = 52 N 2 = 56

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Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
BuyFind

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

Solutions

Chapter
Section
Chapter 8, Problem 8.12P
Textbook Problem
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S W A large counseling center needs to evaluate several experimental programs. Write a paragraph summarizing the results of these tests. Did the new programs work?

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)
P s 1 = 0.53 P s 2 = 0.59
N 1 = 78 N 2 = 82

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)
P s 1 = 0.10 P s 2 = 0.15
N 1 = 52 N 2 = 56

Expert Solution
To determine

(a)

To find:

The significant difference between two sample proportions.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1
(Special Program)
Sample 2
(Regular Program)
Ps1=0.53 Ps2=0.59
N1=78 N2=82

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,

Z(obtained)=(Ps1Ps2)(Pu1Pu2)σpp

Where, Ps1 and Ps2 is the proportion of first and second sample respectively,

Pu1 and Pu2 is the proportion of first and second population respectively,

σpp is the standard deviation and the formula to calculate σpp is given by,

σpp=Pu(1Pu)N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

And Ps1 is the population proportion and the formula to Ps1 is given by,

Pu=N1Ps1+N2Ps2N1+N2

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 two-sample 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,

H0:Pu1=Pu2

H1:Pu1>Pu2

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,

α=0.05

Area of critical region is,

Z(critical)=±1.65

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.53 for Ps1, 0.59 for Ps2, 78 for N1, and 82 for N2 in the above mentioned formula,

Pu=78×0

Expert Solution
To determine

(b)

To find:

The significant difference between two sample proportions.

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