# For each problem, test the sample proportions for the significance of the difference. a. Sample 1 Sample 2 P s 1 = 0.20 P s 2 = 0.17 N 1 = 114 N 2 = 101 b. Sample 1 Sample 2 P s 1 = 0.60 P s 2 = 0.62 N 1 = 478 N 2 = 532

### Essentials Of Statistics

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

### Essentials Of Statistics

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

#### Solutions

Chapter
Section
Chapter 8, Problem 8.10P
Textbook Problem
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## For each problem, test the sample proportions for the significance of the difference.a. Sample 1 Sample 2 P s 1 = 0.20 P s 2 = 0.17 N 1 = 114 N 2 = 101 b. Sample 1 Sample 2 P s 1 = 0.60 P s 2 = 0.62 N 1 = 478 N 2 = 532

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 Sample 2 Ps1=0.20 Ps2=0.17 N1=114 N2=101

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, a two 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:Pu1Pu2

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)=±1.96

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate Pu is given by,

Pu=N1Ps1+N2Ps2N1+N2

Substitute 0.20 for Ps1, 0.17 for Ps2, 114 for N1, and 101 for N2 in the above mentioned formula,

Pu=114×0

Expert Solution
To determine

(b)

To find:

The significant difference between two sample proportions.

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