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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 descriptive statistics and zero order correlations. 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 Mean = 63.6 8.2 55.8 s = 5.5 1.7 5.3 Unemployment Rate Negative Ads Turnout 0.95 − 0.87 Unemployment rate − 0.70 a. Compute the partial correlation coefficient for the relationship between turnout ( Y ) and unemployment ( X ) while controlling for the effect of negative advertising ( Z ) . What effect does this control variable have on the bivariate relationship? Is the relationship between turnout and unemployment direct? ( HINT : Use Formula 13.1 ) b . Compute the partial correlation coefficient for the relationship between turnout ( Y ) and negative advertising ( X ) while controlling for the effect of unemployment ( Z ) . What effect does this have on the bivariate relationship? Is the relationship between turnout and negative advertising direct? ( HINT: Use Formula 13. 1 You will need this partial correlation to compute the multiple correlation coefficient.) c. Find the unstandardized multiple regression equation with unemployment ( X 1 ) and negative ads ( X 2 ) as the independent variables. What turnout would be expected in a city in which the unemployment rate was 10% and 75% of the campaign ads were negative? ( HINT: Use Formulas 13.4 and 13.5 to compute the partial slopes and then use Formula 13.6 to find a , the Y intercept. The regression line is stated in formula 13.3. Substitute 10 for X 1 and 75 for X 2 , 10 compute predicted Y .) 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 ( R 2 ) . How much of the variance in voter turnout is explained by the two independent variables combined? ( HINT: Use Formula 13. 11. You calculated r y 2.1 2 . in part b of this problem.) f. Write a paragraph summarizing your conclusions about the relationships among these three variables.

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Essentials Of Statistics

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

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Chapter
Section
BuyFindarrow_forward

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
Chapter 13, Problem 13.1P
Textbook Problem
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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 descriptive statistics and zero order correlations.

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
Mean = 63.6 8.2 55.8
s = 5.5 1.7 5.3
Unemployment
Rate
Negative Ads
Turnout 0.95 0.87
Unemployment rate 0.70

a. Compute the partial correlation coefficient for the relationship between turnout ( Y ) and unemployment ( X ) while controlling for the effect of negative advertising ( Z ) . What effect does this control variable have on the bivariate relationship? Is the relationship between turnout and unemployment direct? (HINT: Use Formula 13.1)

b. Compute the partial correlation coefficient for the relationship between turnout ( Y ) and negative advertising ( X ) while controlling for the effect of unemployment ( Z ) . What effect does this have on the bivariate relationship? Is the relationship between turnout and negative advertising direct? (HINT: Use Formula 13. 1 You will need this partial correlation to compute the multiple correlation coefficient.)

c. Find the unstandardized multiple regression equation with unemployment ( X 1 ) and negative ads ( X 2 ) as the independent variables. What turnout would be expected in a city in which the unemployment rate was 10% and 75% of the campaign ads were negative? (HINT: Use Formulas 13.4 and 13.5 to compute the partial slopes and then use Formula 13.6 to find a , the Y intercept. The regression line is stated in formula 13.3. Substitute 10 for X 1 and 75 for X 2 , 10 compute predicted Y .)

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 ( R 2 ) . How much of the variance in voter turnout is explained by the two independent variables combined? (HINT: Use Formula 13. 11. You calculated r y 2.1 2 . in part b of this problem.)

f. Write a paragraph summarizing your conclusions about the relationships among these three variables.

To determine

(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 (X)
Negative Ads (Z)
Turnout (Y) 0.95 0.87
Unemployment rate (X) 0.70

Formula used:

The formula to calculate the first order partial correlation is given by,

ryx.z=ryx(ryz)(rxz)1ryz21rxz2

Where, rxz, ryz, and ryx are the zero order correlations.

Calculation:

From the given correlation matrix, the zero order correlations are,

ryx=0.95

rxz=0.70

And,

ryz=0.87

Substitute 0.95 for ryx, 0.70 for rxz, and 0.87 for ryz in the above mentioned formula,

ryx.z=0.95(0.87)(0.70)1(0.87)21(0.70)2=0.950.60910.756910.49=0.34100.24310.51=0.3410.4931×0.7141

Simplify further,

ryx.z=0.3410.35210.97

The first order partial correlation ryx.z measures the strength of the relationship between turnout and unemployment while controlling for the effect of negative advertising. It is higher in value than the zero order coefficient ryx=0.95, but the difference of the two is not much. This suggests the direct relationship between variables X and Y.

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.

To determine

(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 0.92 and the relationship between turnout and negative ads is direct.

Explanation of Solution

Given:

The correlation matrix is given in the table below,

Unemployment
Rate (Z)
Negative Ads (X)
Turnout (Y) 0.95 0.87
Unemployment rate (Z) 0.70

Formula used:

The formula to calculate the first order partial correlation is given by,

ryx.z=ryx(ryz)(rxz)1ryz21rxz2

Where, rxz, ryz, and ryx are the zero order correlations.

Calculation:

From the given correlation matrix, the zero order correlations are,

ryx=0.87

rxz=0.70

And,

ryz=0.95

Substitute 0.87 for ryx, 0.70 for rxz, and 0.95 for ryz in the above mentioned formula,

ryx.z=(0.87)(0.95)(0.70)1(0.95)21(0.70)2=0.87+0.66510.902510.49=0.2050.09750.51=0.2050.3122×0.7141

Simplify further,

ryx.z=0.2050.2230.92

The first order partial correlation ryx.z measures the relationship between turnout and negative ads while controlling for the effect of unemployment. It is lesser in value than the zero order coefficient ryx=0.87, but the difference of the two is not much. This suggests the direct relationship between variables X and Y.

Conclusion:

Therefore, the partial correlation coefficient between turnout and negative ads while controlling for the effect of unemployment is 0.92 and the relationship between turnout and negative ads is direct.

To determine

(c)

To find:

The unstandardized multiple regression equation with unemployment (X1) and negative ads (X2) as the independent variable and the expected turnout in a city in which the unemployment rate is 10% and 75% of the campaign was negative.

Answer to Problem 13.1P

Solution:

The unstandardized multiple regression equation with unemployment (X1) and negative ads (X2) and the independent variable is Y=69.324+2.16X1+(0.42)X2, and a turnout of 59.424 is expected in a city in which the unemployment rate is 10% and 75% of the campaign was negative.

Explanation of Solution

Given:

The correlation matrix is given in the table below,

Unemployment
Rate (X1)
Negative Ads (X2)
Turnout (Y) 0.95 0.87
Unemployment rate (X1) 0.70

The descriptive statistics is given in the table below,

Turnout (Y) Unemployment Rate (X1) % Negative Ads (X2)
Mean= 63.6 8.2 55.8
s= 5.5 1.7 5.3

Formula used:

The formula to calculate the partial slopes for the independent variables is given by,

b1=(sys1)(ry1ry2r121r122)

And,

b2=(sys2)(ry2ry1r121r122)

Where, b1 is the partial slope of X1 and Y, b2 is the partial slope of X2 and Y, sy is the standard deviation of Y, s1 is the standard deviation of the first independent variable X1, s2 is the standard deviation of the second independent variable X2, ry1 is the bivariate correlation between Y and X1, ry2 is the bivariate correlation between Y and X2, r12 is the bivariate correlation between X1 and X2.

The least square multiple regression line is given by,

Y=a+b1X1+b2X2

The formula to calculate the Y intercept of the regression line is given by,

a=Y¯b1X¯1b2X¯2

Calculation:

From the given information,

sy=5.5, s1=1.7, s2=5.3, ry1=0.95, ry2=0.87, r12=0.70, Y¯=63.6, X¯1=8.2, and X¯2=55.8.

The formula for partial slopes of unemployment rate is given by,

b1=(sys1)(ry1ry2r121r122)

Substitute 5.5 for sy, 1.7 for s1, 0.95 for ry1, 0.87 for ry2 and 0.70 for r12 in the above mentioned formula,

b1=(5.51.7)(0.95(0.87)(0.70)1(0.70)2)=(3.2353)(0.950.609010.49)=(3.2353)(0.34100.51)=2.16(1)

The formula for partial slopes of negative ads is given by,

b2=(sys2)(ry2ry1r121r122)

Substitute 5.5 for sy, 5.3 for s2, 0.95 for ry1, 0.87 for ry2 and 0.70 for r12 in the above mentioned formula,

b2=(5.55.3)((0.87)(0.95)(0.70)1(0.70)2)=(1.0377)((0.87)+0.665010.49)=(1.0377)(0.20500.51)=0.42(2)

The formula to calculate the Y intercept of the regression line is given by,

a=Y¯b1X¯1b2X¯2

From equation (1) and (2) substitute 63.6 for Y¯, 2.16 for b1, 0.42 for b2, 8.2 for X¯1, 55.8 for X¯2 in the above mentioned formula,

a=63.62.16(8.2)(0.42)55.8=63.617.712+23.436=69.324(3)

The least square multiple regression line is given by,

Y=a+b1X1+b2X2

From equation (1), (2), and (3), substitute 69.324 for a, 2.16 for b1, and 0.42 for b2 in the above mentioned line,

Y=69.324+2.16X1+(0.42)X2

For predicted value of Y, substitute 10 for X1 and 75 for X2 in the above mentioned line,

Y=69.324+2.16(10)+(0.42)(75)=69.324+21.631.5=59.424

Thus, the unstandardized multiple regression equation with unemployment (X1) and negative ads (X2) and the independent variable is Y=69.324+2.16X1+(0.42)X2, and a turnout of 59.424 is expected in a city in which the unemployment rate is 10% and 75% of the campaign were negative.

Conclusion:

Therefore, the unstandardized multiple regression equation with unemployment (X1) and negative ads (X2) and the independent variable is Y=69.324+2.16X1+(0.42)X2, and a turnout of 59.424 is expected in a city in which the unemployment rate is 10% and 75% of the campaign was negative.

To determine

(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 0.4047. The unemployment rate has a stronger impact on turnout than negative advertising.

Explanation of Solution

Given:

The correlation matrix is given in the table below,

Unemployment
Rate (X1)
Negative Ads (X2)
Turnout (Y) 0.95 0.87
Unemployment rate (X1) 0.70

The descriptive statistics is given in the table below,

Turnout (Y) Unemployment Rate (X1) % Negative Ads (X2)
Mean= 63.6 8.2 55.8
s= 5.5 1.7 5.3

Formula used:

The formula to calculate the beta weights for two independent variables is given by,

b1=b1(s1sy)

And,

b2=b2(s2sy)

Where, b1 is the partial slope of X1 and Y, b2 is the partial slope of X2 and Y, sy is the standard deviation of Y, s1 is the standard deviation of the first independent variable X1, s2 is the standard deviation of the second independent variable X2.

Calculation:

From the given information and part (c),

sy=5.5, s1=1.7, s2=5.3, b1=2.16, and b2=0.42.

The formula to calculate the beta weights for unemployment rate is given by,

b1=b1(s1sy)

Substitute 5.5 for sy, 1.7 for s1, and 2.16 for b1 in the above mentioned formula,

b1=2.16(1.75.5)=2.16×0.3091=0.6676

The formula to calculate the beta weights for negative ads is given by,

b2=b2(s2sy)

Substitute 5.5 for sy, 5.3 for s2, and 0.42 for b2 in the above mentioned formula,

b2=(0.42)(5.35.5)=(0.42)×0.9636=0.4047

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 0.4047. The unemployment rate has a stronger impact on turnout than the negative ads.

To determine

(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 (X1)
Negative Ads (X2)
Turnout (Y) 0.95 0.87
Unemployment rate (X1) 0.70

The descriptive statistics is given in the table below,

Turnout (Y) Unemployment Rate (X1) % Negative Ads (X2)
Mean= 63.6 8.2 55.8
s= 5.5 1.7 5.3

Formula used:

The formula to calculate the coefficient of multiple determination is given by,

R2=ry12+ry2.12(1ry12)

Where, ry12 is the zero order correlation between Y and X1, the quantity squared. ry2.12 is the partial correlation between Y and X2 while controlling for X1, the quantity squared.

Calculation:

From the given information and part (b),

ry12=0.95, and ry2.12=0.92.

The formula to calculate the coefficient of multiple determination is given by,

R2=ry12+ry2.12(1ry12)

Substitute 0.95 for ry1 and 0.92 for ry2.1 in the above mentioned formula,

R2=(0.95)2+(0.92)2(1(0.95)2)=0.9025+0.8464(10.9025)=0.9025+0.8464×0.0975=0.9025+0.0825=0.985

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.

To determine

(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 (X1)
Negative Ads (X2)
Turnout (Y) 0.95 0.87
Unemployment rate (X1) 0.70

The descriptive statistics is given in the table below,

Turnout (Y) Unemployment Rate (X1) % Negative Ads (X2)
Mean= 63.6 8.2 55.8
s= 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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