Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 10.2 | The Null Hypothesis and the Independent-Measures t Statistic 305
■ Calculating the Estimated Standard Error
To develop the formula for s sM1
we consider the following three points.
2M 2
d
1. Each of the two sample means represents it own population mean, but in each case
there is some error.
M 1
approximates μ 1
with some error
M 2
approximates μ 2
with some error
Thus, there are two sources of error.
2. The amount of error associated with each sample mean is measured by the estimated
standard error of M. Using Equation 9.1 (page 269), the estimated standard
error for each sample mean is computed as follows:
For M 1
s M
5Î
s2 1
For M
n 2
s M
1
5Î
s2 2
n 2
3. For the independent-measures t statistic, we want to know the total amount of error
involved in using two sample means to approximate two population means. To do
this, we will find the error from each sample separately and then add the two errors
together. The resulting formula for standard error is
s 5 sM1 2M 2
d Î s2 1
1 s2 2
(10.1)
n 1
n 2
Because the independent-measures t statistic uses two sample means, the formula for the
estimated standard error simply combines the error for the first sample mean and the error
for the second sample mean (Box 10.1).
BOX 10.1 The Variability of Difference Scores
It may seem odd that the independent-measures t
statistic adds together the two sample errors when it
subtracts to find the difference between the two sample
means. The logic behind this apparently unusual
procedure is demonstrated here.
We begin with two populations, I and II (Figure 10.2).
The scores in population I range from a high of 70 to
a low of 50. The scores in population II range from
30 to 20. We will use the range as a measure of how
spread out (variable) each population is:
For population I, the scores cover a range of 20 points.
For population II, the scores cover a range of 10 points.
If we randomly select one score from population I
and one score from population II and compute the
difference between these two scores (X 1
− X 2
), what
range of values is possible for these differences?
To answer this question, we need to find the biggest
possible difference and the smallest possible
difference. Look at Figure 10.2 where the biggest
difference occurs when X 1
= 70 and X 2
= 20. This
is a difference of X 1
− X 2
= 50 points. The smallest
difference occurs when X 1
= 50 and X 2
= 30. This is
a difference of X 1
− X 2
= 20 points. Notice that the
differences go from a high of 50 to a low of 20. This
is a range of 30 points:
range for population I (X 1
scores) = 20 points
range for population II (X 2
scores) = 10 points
range for the differences (X 1
− X 2
) = 30 points
The variability for the difference scores is found
by adding together the variability for each of the two
populations.
In the independent-measures t statistics, we are
computing the variability (standard error) for a
sample mean difference. To compute this value,
we add together the variability for each of the two
sample means.