Research Design in Occupational Education Copyright 1997. James P. Key. Oklahoma State University Except for those materials which are supplied by different departments of the University (ex. IRB, Thesis Handbook) and references used by permission.

MODULE S5 - "t" TEST

We are called on many times to determine if the mean performance of two groups are significantly different. Those two groups might be students, cattle, plants, or other objects. When attempting to determine if the difference between two means is greater than that expected from chance, the "t" test may be the needed statistical technique. If the data is from a normal population and at least ordinal in nature, then we are surer that this is the technique to use. If you wish to generalize to a population, then the samples must be representative.

"t" is the difference between two sample means measured in terms of the standard error of those means, or "t" is a comparison between two groups means which takes into account the differences in group variation and group size of the two groups. The statistical hypothesis for the "t" test is stated as the null hypothesis concerning differences. There is no significant difference in achievement between group 1 and group 2 on the welding test.

Separate variance formula

Use the separate variance formula if:

Pooled Variance Formula

Use the pooled variance formula if:

Correlated Data Formula

If the samples are related (two measures from the same subject or matched pairs), the correlated data formula is used.

In choosing the correct formula, it is fairly easy to determine if the sample sizes are equal. The number of subjects are either the same or they are not.

However, to determine if the variances are homogeneous, use the formula F = s2 (largest) / s2 (smallest). We compare the calculated F value to the F table value at the .05 or .01 level of significance with n1 - 1 and n2 - 1 degrees of freedom.

If the calculated values >= table value, then the variances are not equal; if the calculated value < table value, then the variances are equal.

Example - Calculate the "t" value to test for differences between the achievement of the two samples.

Sample 1                                                  Sample 2

 x1 x2 1 -2 4 1 -4 16 2 -1 1 3 -2 4 3 0 0 5 0 0 4 1 1 7 2 4 5 2 4 9 4 16 15 0 10 25 0 40

= 3                                                                  = 5

*n - 1 used since n < 30

Test for equal sample sizes and homogeneity of variances

n1 = n2 = 5

F = s2 (largest)/s2 (smallest) = 10/2.5 = 4 with 4 and 4 degrees of freedom

F.05 with 4 and 4 degrees of freedom = 6.39

4 < 6.39 so assume s12 = s22

Since sample sizes and variances are equal, either the separate variance formula or the pooled variance formula may be used.

Separate Variance Formula

with 8 degrees of freedom

Pooled Variance Formula

with 8 degrees of freedom

As shown in the above example, the degrees of freedom are calculated differently depending upon whether the n’s and s’s are equal or not. We must check the degrees of freedom corresponding with the formula we use.

To test the hypothesis, we compare the calculated value to the table value for the significance level we have chosen. If the calculated value >= table value, we reject the null hypothesis and conclude the difference is greater than that expected by chance. If the calculated value < table value, we fail to reject the null hypothesis and conclude this amount of difference could have been the result of chance.

In our example, our calculated value was -1.265 with 8 df and the table value for the .01 level with 8 df was + 3.355. Since |-1.265| < |-3.355|, we accept the null hypothesis and conclude that the mean difference in achievement between the two samples was no greater than would be expected by chance.

Assumptions

1. Representative sample (Random)

2. Normal distribution for population

3. At least ordinal measures

SELF ASSESSMENT

1. State the conditions which would help you determine when to use the "t" test.

2. You wish to find out if one group of steers fed a special ration gained weight faster than a similar group of steers on the regular ration. State the statistical hypothesis you would test.

3. Given the following rate of gain data, choose the "t" formula which fits the data.

 Special Ration (x1) Regular Ration (x2) 1 5 2 6 3 7 4 8 5 9

4. Using the data and the formula from problem 3, calculate the "t" value.

5. Calculate the degrees of freedom for problem 3.

6. Test your hypothesis at the .01 level of significance.