Learn moreLearn moreApplied Statistics Handbook

Table of Contents

 


 

Example:  Simple Linear Regression

 

The following data were collected to estimate the correlation between years of formal education and income at age 35 and are the same data used in an earlier example to estimate Pearson r.

 

Susan

Bill

Bob

Tracy

Joan

Education (years)

12

14

16

18

12

Income ($1000)

25

27

32

44

26

 

 

Education

Income

 

 

 

(Years)

($1000)

 

 

Name

X

Y

XY

X2

Susan

12

25

300

144

Bill

14

27

378

196

Bob

16

32

512

256

Tracy

18

44

792

324

Joan

12

26

312

144

=

72

154

2294

1064

=

5

 

 

 

=

14.4

30.8

 

 

 

Determine Regression Line

 

Estimate the slope (b)

 

 

Estimate the y intercept

 

Given x (education) of 15 years, estimate predicted Y

 or an estimated annual income of $32,485 for a person with 15 years of education

 

Determine the Standard Error of the Estimate

 

Standard error of the estimate of Y     

 

Name

a

b

Susan

-9.649

2.809

24.059

.941

.885

Bill

-9.649

2.809

29.677

-2.677

7.166

Bob

-9.649

2.809

35.295

-3.295

10.857

Tracy

-9.649

2.809

40.913

3.087

9.530

Joan

-9.649

2.809

24.059

1.941

3.767

 

 

 

 

32.205

 

                                       

 

Determine the Confidence interval for predicted Y

 

 

where

Degrees of Freedom (df) = 5-2 or 3

Alpha .05, 2-tailed

Based on the t-distribution (see table) tcv = 3.182

 

Confidence interval for predicted Y

Given a 5% chance of error, the estimated income for a person with 15 years of education will be $32,485 plus or minus $10,424 or somewhere between $22,060 and $42,909.

 

Standard error of the slope estimate

To develop confidence intervals or test hypotheses, we need to estimate the standard error of the slope estimate (sb).

 

 

 

 

 

ESS*

 

Name

X

Y

Susan

12

25

0.8

5.8

Bill

14

27

7.3

0.2

Bob

16

32

10.9

2.6

Tracy

18

44

9.6

13.0

Joan

12

26

3.6

5.8

=

5

 

 

 

Mean =

14.4

30.8

 

 

 

 

32.2

27.4

* Error Sum of Squares (ESS)

 

                                            

 

 

Calculating Confidence Interval

          

Where CV= critical value (t-distribution, 2-tailed, .05 alpha, df=n-2) and 2.8 is the point estimate for the slope.

                                 or

 

Interpretation

The slope for education is between 0.8 and 4.8.


Google

 


Copyright 2015, AcaStat Software. All Rights Reserved.