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S O C Occupational prestige scores for a sample of fathers and their oldest son and oldest daughter are presented in the following table. Family Father’s Prestige Son’s Prestige Daughter’s Prestige A 80 85 82 B 78 80 77 C 75 70 68 D 70 75 77 E 69 72 60 F 66 60 52 G 64 48 48 H 52 55 57 Analyze the relationship between father’s and son’s prestige and the relationship between father’s and daughter’s prestige. For each relationship, a. Compute the slope ( b ) and find the Y intercept ( a ). b. State the least-square regression line. What prestige score would you predict for a son whose father had a prestige score of 72? What prestige score would you predict for a daughter whose father has a prestige score of 72? c. Compute r and r 2 . d. Describe the strength and direction of each relationship in a sentence or two. Does the occupational prestige of the father have an impact on his children? Does it have the same impact for daughters as it does for sons?

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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 12.2P
Textbook Problem
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S O C Occupational prestige scores for a sample of fathers and their oldest son and oldest daughter are presented in the following table.

Family Father’s Prestige Son’s Prestige Daughter’s Prestige
A 80 85 82
B 78 80 77
C 75 70 68
D 70 75 77
E 69 72 60
F 66 60 52
G 64 48 48
H 52 55 57

Analyze the relationship between father’s and son’s prestige and the relationship between father’s and daughter’s prestige. For each relationship,

a. Compute the slope (b) and find the Y intercept (a).

b. State the least-square regression line. What prestige score would you predict for a son whose father had a prestige score of 72? What prestige score would you predict for a daughter whose father has a prestige score of 72?

c. Compute r and r 2 .

d. Describe the strength and direction of each relationship in a sentence or two. Does the occupational prestige of the father have an impact on his children? Does it have the same impact for daughters as it does for sons?

To determine

(a)

To find:

The slope and Y intercept.

Explanation of Solution

Given:

The Occupational prestige scores for a sample of fathers and their oldest son and oldest daughter are presented in the following table.

Family Father’s prestige Son’s prestige Daughter’s prestige
A 80 85 82
B 78 80 77
C 75 70 68
D 70 75 77
E 69 72 60
F 66 60 52
G 64 48 48
H 52 55 57

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 father’s prestige is the dependent variable. Thus, it is represented as Y.

The daughter’s prestige is the independent variables. Thus, it is as X1.

Consider the following table of sum of squares of son’s prestige and father’s prestige.

Family Son’s prestige (X1X¯) (X1X¯)2 Father’s prestige (Y) (YY¯) (X1X¯)(YY¯)
A 85 16.875 284.766 80 10.75 181.406
B 80 11.875 141.016 78 8.75 103.906
C 70 1.875 3.51563 75 5.75 10.7813
D 75 6.875 47.2656 70 0.75 5.15625
E 72 3.875 15.0156 69 0.25 -0.9688
F 60 8.125 66.0156 66 3.25 26.4063
G 48 20.125 405.016 64 5.25 105.656
H 55 13.125 172.266 52 17.25 226.406
Totals 545 (XX¯)2=1134.88 554 (XX¯)(YY¯)=658.75

The value of X1¯ is 68.125 and Y¯ is 69.25.

From above table, substitute 85 for X1 and 68.125 for X1¯ in (X1X1¯).

(X1X1¯)=(8568.125)=16.875

Square the both sides of the equation.

(X1X1¯)2=(16.875)2=284.766

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=284.766+141.016+......+172.266=1134.88

Substitute 658.75 for (X1X¯)(YY¯) and 1134

To determine

(b)

To find:

The least regression line and prestige score.

To determine

(c)

To find:

The coefficients r and r2.

To determine

(d)

To explain:

The strength, direction and impact of father’s occupational prestige on his children.

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