Statistical Methods for Psychology

Having found the mean and the variance of a set of differences between means, we

know most of what we need to know. The general form of the sampling distribution of

mean differences is presented in Figure 7.9.

The final point to be made about this distribution concerns its shape. An important the-

orem in statistics states that the sum or difference of two independent normally distributed

variables is itself normally distributed. Because Figure 7.9 represents the difference be-

tween two sampling distributions of the mean, and because we know that the sampling dis-

tribution of means is at least approximately normal for reasonable sample sizes, the

distribution in Figure 7.9 must itself be at least approximately normal.

The t Statistic

Given the information we now have about the sampling distribution of mean differences,

we can proceed to develop the appropriate test procedure. Assume for the moment that

knowledge of the population variances ( ) is not a problem. We have earlier defined zas a

statistic (a point on the distribution) minus the mean of the distribution, divided by the stan-

dard error of the distribution. Our statistic in the present case is ( ), the observed

difference between the sample means. The mean of the sampling distribution is ( ),

and, as we saw, the standard error of differences between means^10 is

Thus we can write

The critical value for a 5 .05 is z5 61.96 (two-tailed), as it was for the one-sample tests

discussed earlier.

The preceding formula is not particularly useful except for the purpose of showing the

origin of the appropriate t test, since we rarely know the necessary population variances.

=

(X 12 X 2 ) 2 (m 1 2m 2 )

B

s^21

n 1

1

s^22

n 2

z=

(X 12 X 2 ) 2 (m 1 2m 2 )

sX 12 X 2

sX 12 X 2 = 3 sX^21 1sX^22 =

B

s^21

n 1

1

s^22

n 2

m 1 2m 2

X 12 X 2

s^2 i

Section 7.5 Hypothesis Tests Applied to Means—Two Independent Samples 205

X 1 – X 2

+

12

n 1

22

n 2

1 – 2

Figure 7.9 Sampling distribution of mean differences

(^10) Remember that the standard deviation of any sampling distribution is called the standard error of that distribution.
standard error of
differences
between means