# S O C A researcher is studying changes in the student body at her university and has selected a random sample of 163 from the freshman class. The table below compares their characteristics to the student body as a whole. Which differences are significant? Variable Freshmen All Students a. Proportion Republican P s = 0.44 P u = 0.49 b. Proportion majoring in math and science P s = 0.17 P u = 0.09 c. Average family income X ¯ = 63 , 542 s = 1568 μ = 59 , 235 d. Average number of siblings X ¯ = 0.9 s = 0.2 μ = 1.2 e. Average number of telephone calls to parents per week X ¯ = 1.5 s = 0.4 μ = 0.4 f. Average number of close friends living on campus X ¯ = 4.5 s = 0.3 μ = 7.8

### 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 7, Problem 7.19P
Textbook Problem
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## S O C A researcher is studying changes in the student body at her university and has selected a random sample of 163 from the freshman class. The table below compares their characteristics to the student body as a whole. Which differences are significant? Variable Freshmen All Students a. Proportion Republican P s = 0.44 P u = 0.49 b. Proportion majoring in math and science P s = 0.17 P u = 0.09 c. Average family income X ¯ = 63 , 542 s = 1568 μ = 59 , 235 d. Average number of siblings X ¯ = 0.9 s = 0.2 μ = 1.2 e. Average number of telephone calls to parents per week X ¯ = 1.5 s = 0.4 μ = 0.4 f. Average number of close friends living on campus X ¯ = 4.5 s = 0.3 μ = 7.8

Expert Solution
To determine

a)

To find:

The significant difference between the given statements.

### Explanation of Solution

Given:

The given information is,

A researcher is studying changes in the student body at her university and has selected a random sample of 163 from the freshman class.

The given table of information is,

 Variable Freshmen All Students a. Proportion Republican Ps=0.44 Pu=0.49

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:

For large samples with single sample proportions, the Z value is given by,

Z(obtained)=PsPuPu(1Pu)/N

Where, Ps is the sample proportion,

Pu is the population proportion and,

N is the sample size.

Calculation:

From the given information, the sample size is 163, sample proportion is 0.44 and the population proportion is 0.49.

Follow the steps for two-tailed hypothesis testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Random sampling.

Level of measurement is interval ratio.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

The statement of the null hypothesis is that the proportion of republican from the sample of freshmen is not significantly different from all the students.

Thus, the null and the alternative hypotheses are,

H0:Pu=0.49

H1:Pu0.49

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample is large, Z distribution can be used.

The information given is in the form of proportions, the test for proportion can be used.

Thus, the sampling distribution is Z distribution

Expert Solution
To determine

b)

To find:

The significant difference between the given statements.

Expert Solution
To determine

c)

To find:

The significant difference between the given statements.

Expert Solution
To determine

d)

To find:

The significant difference between the given statements.

Expert Solution
To determine

e)

To find:

The significant difference between the given statements.

Expert Solution
To determine

f)

To find:

The significant difference between the given statements.

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