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P S Problem 12.4 presented data on ten precincts. The information is reproduced here. 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 Take voter turnout as the dependent variable and do the following: a. Find the multiple regression equations (unstandardized). b. What turnout would you expect for a precinct in which 0% of the voters were Democrats and 5% were minorities? c. Compute beta-weights for each independent variable and compare their relative effect on turnout. Which was the more important factor? d. Compute R 2 . e. Write a paragraph summarizing your findings. 12.4 P S 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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Section
BuyFindarrow_forward

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
Chapter 13, Problem 13.2P
Textbook Problem
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P S Problem 12.4 presented data on ten precincts. The information is reproduced here.

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

Take voter turnout as the dependent variable and do the following:

a. Find the multiple regression equations (unstandardized).

b. What turnout would you expect for a precinct in which 0% of the voters were Democrats and 5% were minorities?

c. Compute beta-weights for each independent variable and compare their relative effect on turnout. Which was the more important factor?

d. Compute R 2 .

e. Write a paragraph summarizing your findings.

12.4 P S 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

(a)

To find:

The unstandardized multiple regression equation with voter turnout as the dependent variable.

Explanation of Solution

Given:

The table for a sample of ten precincts during the last national election is given below:

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

Formula used:

Let the data values be Xi’s.

The formula to calculate mean is given by,

X¯=i=1NXiN

The formula to calculate standard deviation is given by,

s=(XiX¯)2N

Where, N is the population size.

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

rxy=(XiX¯)(YiY¯)[(XiX¯)2][(YiY¯)2]

Where, X and Y are the two variables.

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,

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.

The mean is given by,

X¯=i=1NXiN

For democrat, substitute 10 for N, 33 for X1, 78 for X2 and so on in the above mentioned formula,

X¯=33+78+...+13+3310=48010=48

The mean of democrats is 48.

For turnout, substitute 10 for N, 36 for X1, 60 for X2 and so on in the above mentioned formula,

Y¯=36+60+...+89+8810=56710=56.7

The mean of turnout is 56.7.

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

Precinct Democrat (Xi) (XiX¯) (XiX¯)2 Voters Turnout (Yi) (YiY¯) (YiY¯)2 (XiX¯)(YiY¯)
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 (XiX¯)2=4746 567 (YiY¯)2=3762.1 (XiX¯)(YiY¯)=1957

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 (XiX¯)2 for all the 1iN for the rest data and refer table for the rest of the (XiX¯)2 values calculated. Then the value of (XiX¯)2 is calculated as,

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

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

rxy=(XiX¯)(YiY¯)[(XiX¯)2][(YiY¯)2]

Substitute 1957 for (XiX¯)(YiY¯), 4746 for (XiX¯)2 and 3762.1 for (YiY¯)2 in the above mentioned formula.

rxy=1957(4746)(3762.1)rxy=0.46

The correlation coefficient between democrats and turnouts is 0.46.

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

r2xy=(0.46)2r2xy=0.2116

The formula to calculate standard deviation is given by,

s=(XiX¯)2N

For democrats, substitute 4746 for (XiX¯)2 and 10 for N in the above mentioned formula.

s=474610=474.6=21.8

The standard deviation of democrats is 21.8.

For turnouts, substitute 3762.1 for (YiY¯)2 and 10 for N in the above mentioned formula.

s=3762.110=376.21=19.4

The standard deviation of turnouts is 19.4.

For minority, substitute 10 for N, 9 for X1, 15 for X2 and so on in the above mentioned formula,

X¯=9+15+...+910=10610=10.6

The mean of minority is 10.6.

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

Precinct Minority (Xi) (XiX¯) (XiX¯)2 Voters Turnout (Yi) (YiY¯) (YiY¯)2 (XiX¯)(YiY¯)
A 9 1.6 2.56 36 20.7 428.49 33.12
B 15 4
To determine

(b)

To find:

The expected turnout for a precinct with 0% of democrats voters and 5% of minorities.

To determine

(c)

To find:

The beta weights for each given independent variable and the effect of the variable on turnout.

To determine

(d)

To find:

The coefficient of multiple determination.

To determine

(e)

To explain:

The relationship among the given three variables.

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