Introductory Statistics: Concepts, Models, and Applications
David W. Stockburger

## ONE AND TWO-TAILED t-TESTS

A one- or two-tailed t-test is determined by whether the total area of a is placed in one tail or divided equally between the two tails. The one-tailed t-test is performed if the results are interesting only if they turn out in a particular direction. The two-tailed t-test is performed if the results would be interesting in either direction. The choice of a one- or two-tailed t-test effects the hypothesis testing procedure in a number of different ways.

### TWO-TAILED t-TESTS

A two-tailed t-test divides a in half, placing half in the each tail. The null hypothesis in this case is a particular value, and there are two alternative hypotheses, one positive and one negative. The critical value of t, tcrit, is written with both a plus and minus sign (± ). For example, the critical value of t when there are ten degrees of freedom (df=10) and a is set to .05, is tcrit= ± 2.228. The sampling distribution model used in a two-tailed t-test is illustrated below:

### ONE-TAILED t-TESTS

There are really two different one-tailed t-tests, one for each tail. In a one-tailed t-test, all the area associated with a is placed in either one tail or the other. Selection of the tail depends upon which direction tobs would be (+ or -) if the results of the experiment came out as expected. The selection of the tail must be made before the experiment is conducted and analyzed.

A one-tailed t-test in the positive direction is illustrated below:

The value tcrit would be positive. For example when a is set to .05 with ten degrees of freedom (df=10), tcrit would be equal to +1.812.

A one-tailed t-test in the negative direction is illustrated below:

The value tcrit would be negative. For example, when a is set to .05 with ten degrees of freedom (df=10), tcrit would be equal to -1.812.

### Comparison of One and Two-tailed t-tests

1. If tOBS = 3.37, then significance would be found in the two-tailed and the positive one-tailed t-tests. The one-tailed t-test in the negative direction would not be significant, because was placed in the wrong tail. This is the danger of a one-tailed t-test.

2. If tOBS = -1.92, then significance would only be found in the negative one-tailed t-test. If the correct direction is selected, it can be seen that one is more likely to reject the null hypothesis. The significance test is said to have greater power in this case.

The selection of a one or two-tailed t-test must be made before the experiment is performed. It is not "cricket" to find a that tOBS = -1.92, and then say "I really meant to do a one-tailed t-test." Because reviewers of articles submitted for publication are sometimes suspicious when a one-tailed t-test is done, the recommendation is that if there is any doubt, a two-tailed test should be done.