Statistical Methods for Psychology

It can be shown that bis normally distributed about with a standard error approxi-

mated by^15

Thus, if we wish to test the hypothesis that the true slope of the regression line in the popu-

lation is zero (H 0 : b* 5 0), we can simply form the ratio

which is distributed as t on N 2 2 df.

For our sample data on SAT performance and test-taking ability, b 5 4.865, ,

and

Thus

which is the same answer we obtained when we tested r. Since and

, we will reject and conclude that our regression line has a nonzero

slope. In other words, higher levels of test-taking skills are associated with higher predicted

SAT scores.

From what we know about the sampling distribution of b, it is possible to set up confi-

dence limits on.

where is the two-tailed critical value of ton N 22 df.

For our data the relevant statistics can be obtained from Exhibit 9.1. The 95% confi-

dence limits are

Thus, the chances are 95 out of 100 that the limits constructed in this way will encompass

the true value of. Note that the confidence limits do not include zero. This is in line with

the results of our t test, which rejected

Testing the Difference Between Two Independent bs

This test is less common than the test on a single slope, but the question that it is de-

signed to ask is often a very meaningful one. Suppose we have two sets of data on the re-

lationship between the amount that a person smokes and life expectancy. One set is made

up of females, and the other of males. We have two separate data sets rather than one

large one because we do not want our results to be contaminated by normal differences

H 0 : b*=0.

b*

=4.865 6 3.123=1.742...b*...7.988

CI(b*)=4.865 6 2.056c

53.127

6.73 127

d

ta> 2

CI(b*)=b 6 (ta> 2 )c

(SY#X)

sX 1 N 21

d

b*

t.025(26)=2.056 H 0

tobt=3.202

t=

(4.865)(6.73)( 127 )

53.127

=3.202

sY#X=53.127.

sX=6.73

t=

b 2 b*

sb

=

b

SY#X

sX 1 N 21

=

(b)(sX)( 1 N 21 )

SY#X

sb=

sY#X

sX 1 N 21

b*

Section 9.11 Hypothesis Testing 273

(^15) There is surprising disagreement concerning the best approximation for the standard error of b. Its denominator
is variously given as sX 1 N,sX 1 N 21 ,sX 1 N 22.