Statistical Methods for Psychology

Writing up the Results

If you were writing up the results of this experiment, you might write something like the

following:

This experiment tested the hypothesis that stereotype threat will disrupt the perform-

ance even of a group that is not usually thought of as having a negative stereotype with

respect to performance on math tests. Aronson et al. (1998) asked two groups of partic-

ipants to take a difficult math exam. These were white male college students who re-

ported that they typically performed well in math and that good math performance was

important to them. One group of students (n 5 11) was simply given the math test and

asked to do as well as they could. A second, randomly assigned group (n 5 12), was

informed that Asian males often outperformed white males, and that the test was in-

tended to help to explain the difference in performance. The test itself was the same for

all participants. The results showed that the Control subjects answered a mean of 9.64

problems correctly, whereas the subjects in the Threat group completely only a mean

of 6.58 problems. The standard deviations were 3.17 and 3.03, respectively. This repre-

sents an effect size (d) of .99, meaning that the two groups differed in terms of the

number of items correctly completed by nearly one standard deviation.

Student’s ttest was used to compare the groups. The resulting t(21) was 2.37, and was

significant at p,.05, showing that stereotype threat significantly reduced the performance

of those subjects to whom it was applied. The 95% confidence interval on the difference in

means is 0.3712 m 1 – m 2 5.7488. This is quite a wide interval, but keep in mind that

the two sample sizes were 11 and 12. An alternative way of comparing groups is to note

that the Threat group answered 32% fewer items correctly than did the Control group.

7.7 Heterogeneity of Variance: The Behrens–Fisher Problem

We have already seen that one of the assumptions underlying the t test for two independent

samples is the assumption of homogeneity of variance( ). To be more spe-

cific, we can say that when is true and whenwe have homogeneity of variance, then,

pooling the variances, the ratio

is distributed as t on n 11 n 22 2 df. If we can assume homogeneity of variance there is no dif-

ficulty, and the techniques discussed in this section are not needed. When we do not have ho-

mogeneity of variance, however, this ratio is not, strictly speaking, distributed as t. This leaves

us with a problem, but fortunately a solution (or a number of competing solutions) exists.

First of all, unless , it makes no sense to pool (average) variances be-

cause the reason we were pooling variances in the first place was that we assumed them to

be estimating the same quantity. For the case of heterogeneous variances,we will first

dispense with pooling procedures and define

t¿=

(X 12 X 2 )

D

s^21

n 1

1

s^22

n 2

s^21 =s^22 =s^2

t=

(X 12 X 2 )

D

s^2 p

n 1

1

s^2 p

n 2

H 0

s^21 =s^22 =s^2

... ...

Section 7.7 Heterogeneity of Variance: The Behrens–Fisher Problem 213

homogeneity of
variance

heterogeneous
variances