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P A Is there a relationship between salary levels and unionization for public employees? The following data represent this relationship for fire departments in a random sample of 100 cities of roughly the same size. Salary data have been dichotomized at the median. Explain your results in a sentence or two, citing both the results of the chi square test and the patterns of the column percentages. Salary Status Union Non-Union Totals High 21 29 50 Low Totals 14 35 36 65 50 100

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Essentials Of Statistics

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

Solutions

Chapter
Section
BuyFindarrow_forward

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
Chapter 10, Problem 10.5P
Textbook Problem
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P A Is there a relationship between salary levels and unionization for public employees? The following data represent this relationship for fire departments in a random sample of 100 cities of roughly the same size. Salary data have been dichotomized at the median. Explain your results in a sentence or two, citing both the results of the chi square test and the patterns of the column percentages.

Salary Status
Union Non-Union Totals
High 21 29 50
Low
Totals
14 35 36 65 50 100

To determine

To find:

The chi square value for the given information and test for its significance.

Explanation of Solution

Given:

The given statement is,

The following data represent this relationship for fire departments in a random sample of 100 cities of roughly the same size.

The given table of information is,

Salary Status
Union Non-Union Totals
High 21 29 50
Low 14 36 50
Totals 35 65 100

Approach:

The confidence interval is an interval estimate from the statistics of the observed data that might contain the true value of the unknown population parameter.

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 a chi square, the expected frequency fe is given as,

fe=Rowmarginal×ColumnmarginalN

Where N is the total of frequencies.

The chi square statistic is given by,

χ2(obtained)=(fofe)2fe

Where fo is the observed frequency,

And fe is the expected frequency.

The degrees of freedom for the bivariate table is given as,

df=(r1)(c1)

Where r is the number of rows and c is the number of columns.

Calculation:

From the given information,

The observed frequency is given as,

fe=Rowmarginal×ColumnmarginalN……(1)

Substitute 50 for row marginal, 35 for column marginal and 100 for N in equation (1).

f1=50×35100=17.5

Substitute 50 for row marginal, 65 for column marginal and 100 for N in equation (1).

f2=50×65100=32.5

Substitute 50 for row marginal, 35 for column marginal and 100 for N in equation (1).

f3=50×35100=17.5

Substitute 50 for row marginal, 65 for column marginal and 100 for N in equation (1).

f4=50×65100=32.5

Consider the following table,

6.25 fo fe fofe (fofe)2 (fofe)2/fe
21 17.5 3.5 12.25 0.70
29 32.5 3.5 12.25 0.38
14 17.5 3.5 12.25 0.70
36 32.5 3.5 12.25 0.38
Total 100 100 0 χ2=2

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