Statistical Methods for Psychology

in life expectancy between males and females. Suppose further that we obtained the

following data:

Males Females

b 2 0.40 2 0.20

2.10 2.30

2.50 2.80

N 101 101

It is apparent that for our data the regression line for males is steeper than the regres-

sion line for females. If this difference is significant, it means that males decrease their life

expectancy more than do females for any given increment in the amount they smoke. If this

were true, it would be an important finding, and we are therefore interested in testing the

difference between and.

The t test for differences between two independent regression coefficients is directly anal-

ogous to the test of the difference between two independent means. If is true ( ),

the sampling distribution of is normal with a mean of zero and a standard error of

This means that the ratio

is distributed as t on df. We already know that the standard error of bcan be

estimated by

and therefore can write

where and are the error variances for the two samples. As was the case with

means, if we assume homogeneity of error variances, we can pool these two estimates,

weighting each by its degrees of freedom:

For our data,

Substituting this pooled estimate into the equation, we obtain

=

B

4.85

(2.5)(100)

1

4.85

(2.8)(100)

=0.192

sb 12 b 2 =

C

s^2 Y#X 1

s^2 X 1 (N 12 1)

1

s^2 Y#X 2

s^2 X 2 (N 22 1)

s^2 Y#X=

99(2.10^2 ) 1 99(2.30^2 )

101110124

=4.85

s^2 Y#X=

(N 12 2)s^2 Y#X 11 (N 22 2)s^2 Y#X 2

N 11 N 224

s^2 Y#X 1 s^2 Y#X 2

sb 12 b 2 =

C

s^2 Y#X 1

s^2 X 1 (N 12 1)

1

s^2 Y#X 2

s^2 X 2 (N 22 1)

sb=

sY#X

sX 1 N 21

N 11 N 224

t=

b 12 b 2

3 s^2 b 11 s^2 b 2

sb 12 b 2 = 3 s^2 b 11 s^2 b 2

b 12 b 2

H 0 H 0 : b* 1 =b* 2

b 1 b 2

s^2 X

sY#X

274 Chapter 9 Correlation and Regression