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The following variables were collected for a sample of ten precincts during the last national election. Compute r and r 2 for each combination of variables and write a paragraph interpreting the relationship between these variables. Take voter turnout as the dependent variable. Precinct Percentage Democrat Percentage Minority Percentage Voter Turnout A 33 9 36 B 78 15 60 C 62 18 64 D 50 10 56 E 45 12 55 F 56 8 52 G 85 20 25 H 25 0 42 I 13 5 89 J 33 9 88

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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 12, Problem 12.4P
Textbook Problem
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The following variables were collected for a sample of ten precincts during the last national election. Compute r and r 2 for each combination of variables and write a paragraph interpreting the relationship between these variables. Take voter turnout as the dependent variable.

Precinct Percentage
Democrat
Percentage Minority Percentage Voter Turnout
A 33 9 36
B 78 15 60
C 62 18 64
D 50 10 56
E 45 12 55
F 56 8 52
G 85 20 25
H 25 0 42
I 13 5 89
J 33 9 88

To determine

To find:

The coefficients r and r2.

Explanation of Solution

Given:

The given statement is,

The following variables were collected for a sample of ten precincts during the last national election. The voter turnout is the dependent variable.

The given table of information is,

Precinct Percentage
Democrat
Percentage Minority Percentage Voter Turnout
A 33 9 36
B 78 15 60
C 62 18 64
D 50 10 56
E 45 12 55
F 56 8 52
G 85 20 25
H 25 0 42
I 13 5 89
J 33 9 88

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,

X is the independent variable,

a is the Y intercept of the regression line

and, b is the slope of the regression line.

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

The rest of the variables, i.e. democrat and minority are the independent variables and are represented as X1 and X2 respectively.

Consider the following table of sum of squares of democrat and voter turnout.

Precinct Democrat (X1) (X1X¯) (X1X¯)2 Voters Turnout (Y) (YY¯) (YY¯)2 (X1X¯)(YY¯)
A 33 15 225 36 20.7 428.49 310.5
B 78 30 900 60 3.3 10.89 99
C 62 14 196 64 7.3 53.29 102.2
D 50 2 4 56 0.7 0.49 1.4
E 45 3 9 55 1.7 2.89 5.1
F 56 8 64 52 4.7 22.09 37.6
G 85 37 1369 25 31.7 1004.89 1172.9
H 25 23 529 42 14.7 216.09 338.1
I 13 35 1225 89 32.3 1043.29 1130.5
J 33 15 225 88 31.3 979.69 469.5
Totals 480 (X1X¯)2=4746 567 (YY¯)2=3762.1 (XX¯)(YY¯)=1957

From above table, substitute 33 for X1 and 48 for X¯ in (X1X¯).

(X1X¯)=(3348)(X1X¯)=15

Square the both sides of the equation.

(X1X¯)2=(15)2(X1X¯)2=225

Proceed in the same manner to calculate (X1X¯)2 for all the 1iN for the rest data and refer table for the rest of the (X1X¯)2 values calculated. Then the value of (X1X¯)2 is calculated as,

(X1X¯)2=225++900+196+4+9+64+1369+529+1225+225=4746

Substitute 1957 for (XX¯)(YY¯), 4746 for (X1X¯)2 and 3762

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