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The data sets in the first three problems are unrealistically small to allow you to practice the computational routines presented in this chapter. Most problems use SPSS.
In problem 12.1, data regarding voter turnout in five cities was presented. For the sake of convenience, the data for three of the variables arc presented again here along with
| City | Turnout | Unemployment Rate | % Negative Ads | |
| A | 55 | 5 | 60 | |
| B | 60 | 8 | 63 | |
| C | 65 | 9 | 55 | |
| D | 68 | 9 | 53 | |
| E | 70 | 10 | 48 | |
|
|
63.6 | 8.2 | 55.8 | |
|
|
5.5 | 1.7 | 5.3 | |
| Unemployment Rate |
Negative Ads | |||
| Turnout | 0.95 |
|
||
| Unemployment rate |
|
|||
a. Compute the partial
b. Compute the partial correlation coefficient for the relationship between turnout
c. Find the unstandardized multiple regression equation with unemployment
d. Compute beta-weights for each independent variable. Which has the stronger impact on turnout? (HINT: Use Formulas 13. 7 and 13.8 to calculate the beta-weights.)
e. Compute the coefficient of multiple determination
f. Write a paragraph summarizing your conclusions about the relationships among these three variables.
(a)
To find:
The partial correlation coefficient between turnout and unemployment while controlling for the effect of negative advertising.
Answer to Problem 13.1P
Solution:
The partial correlation coefficient between turnout and unemployment while controlling for the effect of negative advertising is 0.97 and the relationship between turnout and unemployment rate is direct.
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
Formula used:
The formula to calculate the first order partial correlation is given by,
Where,
Calculation:
From the given correlation matrix, the zero order correlations are,
And,
Substitute 0.95 for
Simplify further,
The first order partial correlation
Conclusion:
Therefore, the partial correlation coefficient between turnout and unemployment while controlling for the effect of negative advertising is 0.97 and the relationship between turnout and unemployment rate is direct.
(b)
To find:
The partial correlation coefficient between turnout and negative ads while controlling for the effect of unemployment.
Answer to Problem 13.1P
Solution:
The partial correlation coefficient between turnout and negative ads while controlling for the effect of unemployment is
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
Formula used:
The formula to calculate the first order partial correlation is given by,
Where,
Calculation:
From the given correlation matrix, the zero order correlations are,
And,
Substitute
Simplify further,
The first order partial correlation
Conclusion:
Therefore, the partial correlation coefficient between turnout and negative ads while controlling for the effect of unemployment is
(c)
To find:
The unstandardized multiple regression equation with unemployment
Answer to Problem 13.1P
Solution:
The unstandardized multiple regression equation with unemployment
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
The descriptive statistics is given in the table below,
| Turnout |
Unemployment Rate |
% Negative Ads |
|
| 63.6 | 8.2 | 55.8 | |
| 5.5 | 1.7 | 5.3 |
Formula used:
The formula to calculate the partial slopes for the independent variables is given by,
And,
Where,
The least square multiple regression line is given by,
The formula to calculate the
Calculation:
From the given information,
The formula for partial slopes of unemployment rate is given by,
Substitute 5.5 for
The formula for partial slopes of negative ads is given by,
Substitute 5.5 for
The formula to calculate the
From equation
The least square multiple regression line is given by,
From equation
For predicted value of
Thus, the unstandardized multiple regression equation with unemployment
Conclusion:
Therefore, the unstandardized multiple regression equation with unemployment
(d)
To find:
The beta weights for each given independent variable and the variable with the stronger impact on turnout.
Answer to Problem 13.1P
Solution:
The beta weight for unemployment rate is 0.6676 and for negative ads is
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
The descriptive statistics is given in the table below,
| Turnout |
Unemployment Rate |
% Negative Ads |
|
| 63.6 | 8.2 | 55.8 | |
| 5.5 | 1.7 | 5.3 |
Formula used:
The formula to calculate the beta weights for two independent variables is given by,
And,
Where,
Calculation:
From the given information and part
The formula to calculate the beta weights for unemployment rate is given by,
Substitute 5.5 for
The formula to calculate the beta weights for negative ads is given by,
Substitute 5.5 for
Compare the beta weights, the unemployment rate has the stronger effect than the negative ads on turnouts, the net effect of the first independent variable after controlling the effect of negative ads is positive, while the net effect of second independent variable is negative.
Conclusion:
Therefore, the beta weight for unemployment rate is 0.6676 and for negative ads is
(e)
To find:
The coefficient of multiple determination.
Answer to Problem 13.1P
Solution:
The coefficient of multiple determination is 0.985 and 98.5% of the variance in voter turnout is explained by the two independent variables combined.
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
The descriptive statistics is given in the table below,
| Turnout |
Unemployment Rate |
% Negative Ads |
|
| 63.6 | 8.2 | 55.8 | |
| 5.5 | 1.7 | 5.3 |
Formula used:
The formula to calculate the coefficient of multiple determination is given by,
Where,
Calculation:
From the given information and part
The formula to calculate the coefficient of multiple determination is given by,
Substitute 0.95 for
Thus, 98.5% of the variance in voter turnout is explained by the two independent variables combined.
Conclusion:
Therefore, the coefficient of multiple determination is 0.985 and 98.5% of the variance in voter turnout is explained by the two independent variables combined.
(f)
To explain:
The relationship among the given three variables.
Answer to Problem 13.1P
Solution:
The variation explained between turnout and unemployment rate while controlling for negative advertising is 95%. The variation explained between turnout and negative advertising while controlling for unemployment is in negative 89%, but 98.5% of the variance in voter turnout is explained by unemployment rate and negative ads combined.
Explanation of Solution
Given:
The correlation matrix is given in the table below,
| Unemployment Rate |
Negative Ads |
|
| Turnout |
0.95 | |
| Unemployment rate |
The descriptive statistics is given in the table below,
| Turnout |
Unemployment Rate |
% Negative Ads |
|
| 63.6 | 8.2 | 55.8 | |
| 5.5 | 1.7 | 5.3 |
Approach:
The variation explained between turnout and unemployment rate while controlling for negative advertising is 95%. The variation explained between turnout and negative advertising while controlling for unemployment is in negative 89%, but 98.5% of the variance in voter turnout is explained by unemployment rate and negative ads combined.
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