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G E R The residents of a housing development for senior citizens have completed a survey in which they indicated how physically active they are and how many visitors they receive each week. Are these two variables related for the ten cases reported here? Compute r and r 2 . Find the least-squares regression line. What would be the predicted number of visitors for a person whose level of activity was a 5? How about a person who scored 18 on level of activity? Case Level of Activity Number of Visitors A 10 14 B 11 12 C 12 10 D 10 9 E 15 8 F 9 7 G 7 10 H 3 15 I 10 12 J 9 2

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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 12.3P
Textbook Problem
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G E R The residents of a housing development for senior citizens have completed a survey in which they indicated how physically active they are and how many visitors they receive each week. Are these two variables related for the ten cases reported here? Compute r and r 2 . Find the least-squares regression line. What would be the predicted number of visitors for a person whose level of activity was a 5? How about a person who scored 18 on level of activity?

Case Level of Activity Number of Visitors
A 10 14
B 11 12
C 12 10
D 10 9
E 15 8
F 9 7
G 7 10
H 3 15
I 10 12
J 9 2

To determine

To find:

The coefficients r and r2.

Explanation of Solution

Given:

The given statement is,

The residents of a housing development for senior citizens have completed a survey that indicated their level of inactivity and the number of visitors they receive each week.

The given table of information is,

Case Level of inactivity Number of visitors
A 10 14
B 11 12
C 12 10
D 10 9
E 15 8
F 9 7
G 7 10
H 3 15
I 10 12
J 9 2

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 number of visitors is the dependent variable. Thus, it is represented as Y.

The level of inactivity is the independent variables. Thus, it is as X.

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

Case Level of inactivity (X) (XX¯) (XX¯)2 visitors (Y) (YY¯) (YY¯)2 (XX¯)(YY¯)
A 10 0.4 0.16 14 4.1 16.81 1.64
B 11 1.4 1.96 12 2.1 4.41 2.94
C 12 2.4 5.76 10 0.1 0.01 0.24
D 10 0.4 0.16 9 0.9 0.81 0.36
E 15 5.4 29.16 8 1.9 3.61 10.26
F 9 0.6 0.36 7 2.9 8.41 1.74
G 7 2.6 6.76 10 0.1 0.01 0.26
H 3 6.6 43.56 15 5.1 26.01 33.66
I 10 0.4 0.16 12 2.1 4.41 0.84
J 9 0.6 0.36 2 -7.9 62.41 4.74
Totals 96 (XX¯)2=88

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