State the assumptions for computing a confidence interval on the difference
between means

Compute a confidence interval on the difference between means

Format data for computer analysis

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. We take as an example the data
from the "Animal
Research" case study. In this experiment, students rated
(on a 7-point scale) whether they thought animal research is wrong.
The sample sizes, means, and variances are shown separately for
males and females in Table 1.

Table 1. Means and Variances in Animal Research study.

Condition

n

Mean

Variance

Females

17

5.353

2.743

Males

17

3.882

2.985

As you can see, the females rated animal research
as more wrong than did the males. This sample difference between
the female mean of 5.35 and the male mean of 3.88 is 1.47. However,
the gender difference in this particular sample is not very important.
What is important is the difference in the population.
The difference in sample means is used to estimate the difference
in population means. The accuracy of the estimate is revealed
by a confidence
interval.

In order to construct a confidence interval, we
are going to make three assumptions:

The two populations have the same variance. This assumption
is called the assumption of homogeneity of
variance.

Each value is sampled independently
from each other value.

The consequences of violating these assumptions are discussed in
a later section. For now, suffice it to say that small-to-moderate
violations of assumptions 1 and 2 do not make much difference.

A confidence interval on the difference between
means is computed using the following formula:

where M_{1} - M_{2}
is the difference between sample means, t_{CL}
is the t for the desired level of confidence, and
is the estimated standard
error of the difference between sample means. The meanings
of these terms will be made clearer as the calculations are demonstrated.

We continue to use the data from the "Animal
Research" case study and will compute a confidence interval
on the difference between the mean score of the females and the
mean score of the males. For this calculation, we will assume
that the variances in each of the two populations are equal.

The first step is to compute the estimate of the
standard error of the difference between means ().
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
population is:

In order to estimate this quantity, we estimate
σ^{2} and use that estimate in place
of σ^{2}. 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:

where MSE is our estimate of σ^{2}.
In this example,

MSE = (2.743 + 2.985)/2 = 2.864.

Note that MSE stands for "mean square error" and is the mean squared deviation of each score from its group's mean.

Since n (the number of scores in
each condition) is 17,

==
= 0.5805.

The next step is to find the t to use for the
confidence interval (t_{CL}). To calculate
t_{CL}, we need to know the degrees
of freedom. The degrees of freedom is the number of
independent estimates of variance on which MSE is based. This
is equal to (n_{1} - 1) + (n_{2}
- 1) where n_{1 }is the sample size of the
first group and n_{2} is the sample size
of the second group. For this example, n_{1}=
n_{2} = 17. When n_{1}=
n_{2}, it is conventional to use "n"
to refer to the sample size of each group. Therefore, the degrees
of freedom is 16 + 16 = 32.

From either the above calculator or a t table, you can find that
the t for a 95% confidence interval for 32 df is 2.037.

We now have all the components needed to compute
the confidence interval. First, we know the difference between
means:

M_{1} - M_{2}
= 5.353 - 3.882 = 1.471

We know the standard error of the difference between
means is

= 0.5805

and that the t for the 95% confidence interval
with 32 df is

t_{CL = 2.037}

Therefore, the 95% confidence interval is

Lower Limit = 1.471 - (2.037)(0.5805) = 0.29

Upper Limit = 1.471 + (2.037)(0.5805) = 2.65

We can write the confidence interval as:

0.29 ≤ μ_{f} - μ_{m}
≤ 2.65

where μ_{f} is the population mean for
females and μ_{m} is the population mean
for males. This analysis provides evidence that the mean for females
is higher than the mean for males, and that the difference between
means in the population is likely to be between 0.29 and 2.65.

Formatting data for Computer Analysis

Most computer programs that compute t tests require
your data to be in a specific form. Consider the data in Table 2.

Table 2. Example Data.

Group 1

Group 2

3

5

4

6

5

7

Here there are two groups, each with three observations. To format
these data for a computer program, you normally have to use two
variables: the first specifies the group the subject is in and the
second is the score itself. For the data in Table 2, the reformatted
data look as follows:

Table 3. Reformatted Data.

G

Y

1

3

1

4

1

5

2

5

2

6

2

7

To use Analysis
Lab to do the calculations, you would copy the data and then

Click the "Enter/Edit User Data" button. (You may
be warned that for security reasons you must use the keyboard
shortcut for pasting data.)

Paste your data.

Click "Accept Data."

Set the Dependent Variable to Y.

Set the Grouping Variable to G.

Click the t-test confidence interval button.

The 95% confidence interval on the difference between means extends
from -4.267 to 0.267.

Computations for Unequal Sample Sizes (optional)

The calculations are somewhat more
complicated when the sample sizes are not equal. One consideration
is that MSE, the estimate of variance, counts the sample with
the larger sample size more than the sample with the smaller sample
size. Computationally this is done by computing the sum of squares
error (SSE) as follows:

where M_{1} is the mean for group 1 and
M_{2} is the mean for group 2. Consider
the following small example: