Question

List the requirements for testing the differences between means

Answer 1

Objectives

1. State the assumptions for testing the difference between two means
2. Estimate the population variance assuming homogeneity of variance
3. Compute the standard error of the difference between means
4. Compute t and p for the difference between means
5. Format data for computer analysis

We take as an example the data from the "Animal Research" case study. In this experiment, students rated 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.

Group	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 whether there is a difference in the population means.

In order to test whether there is a difference between population means, we are going to make three assumptions:

1. The two populations have the same variance. This assumption is called the assumption of homogeneity of variance.
2. The populations are normally distributed.
3. Each value is sampled independently from each other value. This assumption requires that each subject provide only one value.

In this case, our statistic is the difference between sample means and our hypothesized value is 0. The hypothesized value is the null hypothesis that the difference between population means is 0.

We continue to use the data from the "Animal Research" case study and will compute a significance test on the difference between the mean score of the females and the mean score of the males. For this calculation, we will make the three assumptions specified above.

The first step is to compute the statistic, which is simply the difference between means.

M₁ - M₂ = 5.3529 - 3.8824 = 1.4705

Since the hypothesized value is 0, we do not need to subtract it from the statistic.

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 (
SM-M2
). the formula for the standard error of the difference between means is:


202 OM- = o o o o? Yn n vnn Vn - + - = - - =

In order to estimate this quantity, we estimate σ² and use that estimate in place of σ². Since we are assuming the two 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:

5+2 MSE

where MSE is our estimate of σ². In this example,

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

Since n is 17,

SM-M2
=
2 MSE
=
(2)(2.864) V 17
= 0.5805.

The next step is to compute t by plugging these values into the formula:

t = 1.4705/.5805 = 2.533.

Finally, we compute the probability of getting a t as large or larger than 2.533 or as small or smaller than -2.533. To do this, 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 (n₁ - 1) + (n₂ - 1), where n₁ is the sample size of the first group and n₂ is the sample size of the second group. For this example, n₁ = n₂ = 17. When n₁ = n₂, it is conventional to use "n" to refer to the sample size of each group. Therefore, the degrees of freedom is 16 + 16 = 32.

Once we have the degrees of freedom, we can use the t distribution calculator to find the probability. the probability value for a two-tailed test is 0.0164.

For one tailed test the probability value of 0.0082 is half the value for the two-tailed test.