Why does voter turnout vary from election to election? For municipal elections in five different cities, information has been gathered on the percentage of citizens who voted (the dependent variable) and three different independent variables: unemployment rate, average years of education for the city, and the percentage of all political ads that used "negative campaigning” (personal attacks, negative portrayals of the opponent’s record, etc.). For each relationship between turnout (Y) and the independent variables, a. Compute the slope ( b ) and find the Y intercept ( a ). ( HINT: Remember to compute b before computing a. A computing table such as table 12.3 is highly recommended. ) b. State the least-square regression line and predict the voter turnout for a city in which the unemployment rate was 12%, a city in which the average years of schooling was 11%, and an election in which 90% of the ads were negative. c. Compute r and r 2 . ( HINT: A computing table such as Table 12.4 is highly recommended. If you constructed one for computing b, you already have most of the quantities you will need to solve for r.) d. Describe the strength and direction of each relationship in a sentence or two. Which of the independent variables had the strongest effect on turnout? City Turnout ( Y ) Unemployment Rate ( X 1 ) Average Years of School ( X 2 ) Percentage of Negative Ads ( X 3 ) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

Essentials Of Statistics

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

Chapter
Section

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
Chapter 12, Problem 12.1P
Textbook Problem
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Why does voter turnout vary from election to election? For municipal elections in five different cities, information has been gathered on the percentage of citizens who voted (the dependent variable) and three different independent variables: unemployment rate, average years of education for the city, and the percentage of all political ads that used "negative campaigning” (personal attacks, negative portrayals of the opponent’s record, etc.). For each relationship between turnout (Y) and the independent variables,a. Compute the slope (b) and find the Y intercept (a). (HINT: Remember to compute b before computing a. A computing table such as table 12.3 is highly recommended.)b. State the least-square regression line and predict the voter turnout for a city in which the unemployment rate was 12%, a city in which the average years of schooling was 11%, and an election in which 90% of the ads were negative.c. Compute r and r 2 . (HINT: A computing table such as Table 12.4 is highly recommended. If you constructed one for computing b, you already have most of the quantities you will need to solve for r.)d. Describe the strength and direction of each relationship in a sentence or two. Which of the independent variables had the strongest effect on turnout? City Turnout(Y) UnemploymentRate ( X 1 ) AverageYears ofSchool ( X 2 ) Percentageof Negative Ads ( X 3 ) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

To determine

(a)

To find:

The slope and Y intercept.

Solution:

The values of intercept are 39, 94.77 and 113.8

The values of slope are 3, 12.67 and 0.9.

Explanation of Solution

Given:

The three different independent variables: unemployment rate, average years of education for the unemployment rate, average years of school and the percentage of all political ads that used negative campaigning is given table below.

 City Turnout(Y) UnemploymentRate (X1) AverageYears ofSchool (X2) Percentageof Negative Ads (X3) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

Approach:

The Y intercept (a) is the point of intersection of the regression line and the y axis.

The slope (b) of the regression line is the amount of change in the dependent variable (Y) by a unit change in the independent variable (X).

If the two variables are unrelated, the regression line would be parallel to the x axis.

Formula used:

The model for the least square regression line is defined as,

Y=a+bX

Where Y is the dependent variable,

b=(XX¯)(YY¯)(XX¯)2

Where Y is the dependent variable and X is the independent variable,

b is the slope of the regression line.

And X¯ and Y¯ are their respective means.

The Y intercept (a) is given by,

a=Y¯bX¯

Where, X¯ is the means of X, Y¯ is the means of Y and b is the slope.

Calculation:

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The Unemployment rate is the independent variables. Thus, it is as X1.

Consider the following table of sum of squares of turnout and Unemployment rate.

 City UnemploymentRate (X1) (X1−X1¯) (X1−X1¯)2 Turnout(Y) (Y−Y¯) (X1−X1¯)(Y−Y¯) A 5 −3.2 10.24 55 −8.6 27.52 B 8 −0.2 0.04 60 −3.6 0.72 C 9 0.8 0.64 65 1.4 1.12 D 9 0.8 0.64 68 4.4 3.52 E 10 1.8 3.24 70 6.4 11.52 Total 41 ∑(X1−X¯)2=14.8 318 ∑(X1−X¯)(Y−Y¯)=44.4

The value of X1¯ is 8.2 and Y¯ is 63.6.

From above table, substitute 5 for X1 and 8.2 for X1¯ in (X1X1¯).

(X1X1¯)=(58.2)=3.2

Square the both sides of the equation.

(X1X1¯)2=(3.2)2=10.24

Proceed in the same manner to calculate (X1X1¯)2 for all the 1iN for the rest data and refer table for the rest of the (X1X1¯)2 values calculated. Then the value of (X1X1¯)2 is calculated as,

(X1X1¯)2=10.24+0.04+0.64+0.64+3.24=14.8

Substitute 44.4 for (X1X¯1)(YY¯) and 14.8 for (X1X1¯)2 in the above mentioned formula.

b1=44.414.8=3

Substitute 8.2 for X1¯, 63.6 for Y¯ and 3 for b in the above mentioned formula to solve for intercept.

a1=63.63(8.2)=39

The value of intercept is 39 and the value of slope is 3.

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The average years of school is the independent variables. Thus, it is as X2.

Consider the following table of sum of squares of turnout and Average years of school.

 City AverageYears ofSchool (X2) (X2−X2¯) (X2−X2¯)2 Turnout(Y) (Y−Y¯) (X2−X2¯)(Y−Y¯) A 11.9 −0.6 0.36 55 −8.6 5.16 B 12.1 −0.4 0.16 60 −3.6 1.44 C 12.7 0.2 0.04 65 1.4 0.28 D 12.8 0.3 0.09 68 4.4 1.32 E 13.0 0.5 0.25 70 6.4 3.2 Total 62.5 ∑(X2−X2¯)2=0.9 318 ∑(X2−X¯2)(Y−Y¯)=11.4

The value of X2¯ is 12.5 and Y¯ is 63.6.

From above table, substitute 11.9 for X2 and 12.5 for X2¯ in (X2X2¯).

(X2X2¯)=(11.912.5)=0.6

Square the both sides of the equation.

(X2X2¯)=(0.6)2=0.36

Proceed in the same manner to calculate (X2X2¯)2 for all the 1iN for the rest data and refer table for the rest of the (X2X2¯)2 values calculated. Then the value of (X2X2¯)2 is calculated as,

(X2X2¯)2=0.36+0.16+0.04+0.09+0.25=0.9

Substitute 11.4 for (X2X¯2)(YY¯) and 0.9 for (X2X2¯)2 in the above mentioned formula.

b2=11.40.9=12.67

Substitute 12.5 for X¯, 63.6 for Y¯ and 12.67 for b in the above mentioned formula to solve for intercept.

a2=63.612.67(12.5)=94.77

The value of intercept is 94.77 and the value of slope is 12.67.

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The percentage of negative ads is the independent variables. Thus, it is as X3.

Consider the following table of sum of squares of turnout and Average years of school.

 City Percentageof Negative Ads (X3) (X3−X3¯) (X3−X3¯)2 Turnout(Y) (Y−Y¯) (X3−X3¯)(Y−Y¯) A 60 4.2 17.64 55 −8.6 −36.12 B 63 7.2 51.84 60 −3.6 −25.92 C 55 −0.8 0.64 65 1.4 −1.12 D 53 −2.8 7.84 68 4.4 −12.32 E 48 −7.8 60.84 70 6.4 −49.92 Total 279 ∑(X3−X3¯)2=138.8 318 ∑(X3−X¯3)(Y−Y¯)=−125.4

The value of X3¯ is 55.8 and Y¯ is 63.6.

From above table, substitute 60 for X3 and 55.8 for X3¯ in (X3X3¯).

(X3X3¯)=(6055.8)=4.2

Square the both sides of the equation.

(X3X3¯)2=(4.2)2=17.64

Proceed in the same manner to calculate (X3X3¯)2 for all the 1iN for the rest data and refer table for the rest of the (X3X3¯)2 values calculated. Then the value of (X3X3¯)2 is calculated as,

(X3X3¯)2=36.12+(25.92)+(1.12)+(12.32)+(49.92)=125.4

Substitute 125.4 for (X3X¯3)(YY¯) and 138.8 for (X3X3¯)2 in the above mentioned formula.

b2=125.4138.8=0.9

Substitute 55.8 for X¯, 63.6 for Y¯ and 0.9 for b in the above mentioned formula to solve for intercept.

a2=63.6(0.9)(55.8)=113.82

The value of intercept is 113.82 and the value of slope is 0.9.

Conclusion:

The values of intercept are 39, 94.77 and 113.8

The values of slope are 3, 12.67 and 0.9.

To determine

(b)

To find:

The least regression line and prestige score.

Solution:

The least square regression line for unemployment rate and turnout is,

Y1=39+3X1

The least square regression line for average years of school and turnout model is,

Y2=94.77+12.67X2

The least square regression line for percentage of negative ads and turnout model is,

Y3=113.80.9X3

The voter turnout is 75, 44.6 and 32.82.

Explanation of Solution

Given:

The three different independent variables: unemployment rate, average years of education for the unemployment rate, average years of school and the percentage of all political ads that used negative campaigning is given table below.

 City Turnout(Y) UnemploymentRate (X1) AverageYears ofSchool (X2) Percentageof Negative Ads (X3) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

Approach:

The Y intercept (a) is the point of intersection of the regression line and the y axis.

The slope (b) of the regression line is the amount of change in the dependent variable (Y) by a unit change in the independent variable (X).

If the two variables are unrelated, the regression line would be parallel to the x axis.

Formula used:

The model for the least square regression line is defined as,

Y=a+bX

Where Y is the dependent variable, a is the intercept of Y and b is the slope.

Calculation:

From the sub-part (a),

The values of intercept are 39, 94.77 and 113.8

The values of slope are 3, 12.67 and 0.9.

Substitute 39 for a and 3 for b in the above mentioned linear regression formula.

The least square regression line for unemployment rate and turnout is,

Y1=39+3X1....(1)

The least square regression line for average years of school and turnout model is,

Y2=94.77+12.67X2......(2)

The least square regression line for percentage of negative ads and turnout model is,

Y3=113.80.9X3....(3)

Substitute 12 for X1 in equation (1) and solve for the voter turnout score.

Y1=39+3(12)=75

Substitute 11 for X2 in equation (2) and solve for the voter turnout score.

Y2=94.77+12.67(11)=44.6

Substitute 90 for X3 in equation (3) and solve for the voter turnout score.

Y3=113.80.9(90)=32.82

Conclusion:

The least square regression line for unemployment rate and turnout is,

Y1=39+3X1

The least square regression line for average years of school and turnout model is,

Y2=94.77+12.67X2

The least square regression line for percentage of negative ads and turnout model is,

Y3=113.80.9X3

The voter turnout is 75, 44.6 and 32.82.

To determine

(c)

To find:

The coefficients r and r2.

Solution:

The values of intercept are 29.71 and 35.6

The values of slope are 0.58 and 0.52.

Explanation of Solution

Given:

The three different independent variables: unemployment rate, average years of education for the unemployment rate, average years of school and the percentage of all political ads that used negative campaigning is given table below,

 City Turnout(Y) UnemploymentRate (X1) AverageYears ofSchool (X2) Percentageof Negative Ads (X3) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

Approach:

The Y intercept (a) is the point of intersection of the regression line and the y axis.

The slope (b) of the regression line is the amount of change in the dependent variable (Y) by a unit change in the independent variable (X).

If the two variables are unrelated, the regression line would be parallel to the x axis.

Formula used:

The formula for calculating the correlation coefficient r is given as,

r=(XX¯)(YY¯)[(XX¯)2][(YY¯)2]

Where X and Y are the two variables

And X¯ and Y¯ are their respective means.

Calculation:

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The Unemployment rate is the independent variables. Thus, it is as X1.

Consider the following table of sum of squares of turnout and Unemployment rate.

 City UnemploymentRate (X1) (X1−X1¯) (X1−X1¯)2 Turnout(Y) (Y−Y¯) (Y−Y¯)2 (X1−X1¯)(Y−Y¯) A 5 −3.2 10.24 55 −8.6 73.96 27.52 B 8 −0.2 0.04 60 −3.6 12.96 0.72 C 9 0.8 0.64 65 1.4 1.96 1.12 D 9 0.8 0.64 68 4.4 19.36 3.52 E 10 1.8 3.24 70 6.4 40.96 11.52 Total 41 ∑(X1−X¯)2=14.8 318 ∑(Y−Y¯)2=149.2 ∑(X1−X¯)(Y−Y¯)=44.4

The value of X1¯ is 8.2 and Y¯ is 63.6.

From above table, substitute 5 for X1 and 8.2 for X1¯ in (X1X1¯).

(X1X1¯)=(58.2)=3.2

Square the both sides of the equation.

(X1X1¯)2=(3.2)2=10.24

Proceed in the same manner to calculate (X1X1¯)2 for all the 1iN for the rest data and refer table for the rest of the (X1X1¯)2 values calculated. Then the value of (X1X1¯)2 is calculated as,

(X1X1¯)2=10.24+0.04+0.64+0.64+3.24=14.8

Substitute 44.4 for (X1X¯1)(YY¯), 14.8 for (X1X¯)2 and 149.2 for in (Y1Y¯)2 the above mentioned formula.

r1=44.4(14.8)(149.2)=0.94

Square the above calculated value of r and solve for r2.

r12=(0.94)2r12=0.89

The coefficient of r and r2 is 0.94 and 0.89 respectively.

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The average year of school is the independent variables. Thus, it is as X2.

Consider the following table of sum of squares of turnout and Average years of school.

 City AverageYears ofSchool (X2) (X2−X2¯) (X2−X2¯)2 Turnout(Y) (Y−Y¯) (Y−Y¯)2 (X2−X2¯)(Y−Y¯) A 11.9 −0.6 0.36 55 −8.6 73.96 5.16 B 12.1 −0.4 0.16 60 −3.6 12.96 1.44 C 12.7 0.2 0.04 65 1.4 1.96 0.28 D 12.8 0.3 0.09 68 4.4 19.36 1.32 E 13.0 0.5 0.25 70 6.4 40.96 3.2 Total 62.5 ∑(X2−X2¯)2=0.9 318 ∑(Y−Y¯)2=149.2 ∑(X2−X¯2)(Y−Y¯)=11.4

The value of X2¯ is 12.5 and Y¯ is 63.6.

From above table, substitute 11.9 for X2 and 12.5 for X2¯ in (X2X2¯).

(X2X2¯)=(11.912.5)=0.6

Square the both sides of the equation.

(X2X2¯)=(0.6)2=0.36

Proceed in the same manner to calculate (X2X2¯)2 for all the 1iN for the rest data and refer table for the rest of the (X2X2¯)2 values calculated. Then the value of (X2X2¯)2 is calculated as,

(X2X2¯)2=0.36+0.16+0.04+0.09+0.25=0.9

Substitute 11.4 for (X2X¯2)(YY¯), 0.9 for (X2X2¯)2 and 561.5 for in (YY¯)2 the above mentioned formula.

r2=11.4(0.9)(149.5)=0.98

Square the above calculated value of r and solve for r2.

r22=(0.98)2=0.96

The coefficient of r and r2 is 0.98 and 0.96 respectively.

From the given information,

The turnout is the dependent variable. Thus, it is represented as Y.

The percentage of negative ads is the independent variables. Thus, it is as X3.

Consider the following table of sum of squares of turnout and Average years of school.

 City Percentageof Negative Ads (X3) (X3−X3¯) (X3−X3¯)2 Turnout(Y) (Y−Y¯) (Y−Y¯)2 (X3−X3¯)(Y−Y¯) A 60 4.2 17.64 55 −8.6 73.96 −36.12 B 63 7.2 51.84 60 −3.6 12.96 −25.92 C 55 −0.8 0.64 65 1.4 1.96 −1.12 D 53 −2.8 7.84 68 4.4 19.36 −12.32 E 48 −7.8 60.84 70 6.4 40.96 −49.92 Total 279 ∑(X3−X3¯)2=138.8 318 ∑(Y−Y¯)2=149.2 ∑(X3−X¯3)(Y−Y¯)=−125.4

The value of X3¯ is 55.8 and Y¯ is 63.6.

From above table, substitute 60 for X3 and 55.8 for X3¯ in (X3X3¯).

(X3X3¯)=(6055.8)=4.2

Square the both sides of the equation.

(X3X3¯)2=(4.2)2=17.64

Proceed in the same manner to calculate (X3X3¯)2 for all the 1iN for the rest data and refer table for the rest of the (X3X3¯)2 values calculated. Then the value of (X3X3¯)2 is calculated as,

(X3X3¯)2=36.12+(25.92)+(1.12)+(12.32)+(49.92)=125.4

Substitute 125.4 for (X3X¯3)(YY¯), 138.8 for (X3X3¯)2 and 561.5 for in (YY¯)2 the above mentioned formula.

r3=125.4(138.8)(149.5)=0.87

Square the above calculated value of r and solve for r2.

r32=(0.87)2=0.76

The coefficient of r and r2 is 0.87 and 0.76 respectively.

Conclusion:

The coefficients of r are 0.94, 0.98 and 0.87.

The coefficients of r2 are 0.89, 0.96 and 0.76

To determine

(d)

To explain:

The strength, direction and impact of given independent variable on voter’s turnout.

Solution:

The required explanation is stated.

Explanation of Solution

Given:

The three different independent variables: unemployment rate, average years of education for the unemployment rate, average years of school and the percentage of all political ads that used negative campaigning is given table below.

 City Turnout(Y) UnemploymentRate (X1) AverageYears ofSchool (X2) Percentageof Negative Ads (X3) A 55 5 11.9 60 B 60 8 12.1 63 C 65 9 12.7 55 D 68 9 12.8 53 E 70 10 13.0 48

Calculation:

From sub-part (c), the positive value of correlation between unemployment rate and voter’s turnout shows that there is positive relation between unemployment rate and voter turnout.

The coefficient of determination shows that 89% of variation in the voter’s turnout will be explained by unemployment rate.

The positive value of correlation between average years of school and voter’s turnout shows that there is positive relation between average years of school and voter’s turnout.

The coefficient of determination shows that 96% of variation in the voter’s turnout will be explained by average years of school.

The negative value of correlation between negative campaigning and voter’s turnout shows that there is negative relation between negative campaigning and voter’s turnout

The coefficient of determination shows that 76% of variation in the voter’s turnout will be explained by negative campaigning.

Conclusion:

The required explanation is stated.

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