Differences between Two Means (Independent
Distribution of Difference between Means, Confidence
Interval on the Difference between Means, Hypothesis Testing,
Testing a Single Mean
- State the assumptions for testing the difference between two means
- Estimate the population variance assuming homogeneity of variance
- Compute the standard error of the difference between means
- Compute t and p for the difference between means.
It is much more common for a researcher to be
interested in the difference between means than in the specific
values of the means themselves. This section covers how to test
for differences between means from two separate groups of subjects.
A later section describes how to test for differences between
the means of two condition in designs in where only one group
of subjects is used and each subject is tested in each condition.
In order to test whether there is a difference between
population means, we are going to make three assumptions:
- The two populations have the same variance. This assumption
is called the assumption of homogeneity
- The populations are normally
- Each value is sampled independently
from each other value. This assumption requires that each
subject provide only one value. If a subject provides two
scores, then the value are not independent. The analysis of
data with two scores per subject is shown in the section on
t test later in this chapter.
The consequences of violating the former two assumptions
are investigated in the simulation in
the next section. For now, suffice it to say that small-to-moderate
violations of assumptions 1 and 2 do not make much difference.
It is important not to violate assumption 3.
We saw the following general formula for significance
testing in the section on testing a
The first step is to compute the statistic, which
is simply the difference between means.
The next step is to compute the estimate of the
standard error of the statistic. In this case, the statistic is
the difference between means so the estimated standard error of
the statistic is ().
Recall from the relevant
section in the chapter on sampling distributions that the
formula for the standard error of the difference in means in the
In order to estimate this quantity, we estimate
σ2 and use that estimate in place
of σ2. Since we are assuming the
population variances are the same, we estimate this variance by
averaging our two sample variances. Thus, our estimate of variance
is computed using the following formula:
The next step is to compute t by plugging these
values into the formula.
Finally, we compute the probability of getting
a t as large or larger than the computed t or as small or smaller
than that. To do this, we need to know
of freedom. The degrees of freedom is the number of
independent estimates of variance on which MSE is based. This
is equal to (n1 -1) + (n2
-1) where n1 is the sample size for the
first group and n2 is the sample size
of the second group.
Once we have the degrees of freedom, we can use
the t distribution calculator to find the probability.
Calculator: t distribution