Introductory Statistics: Concepts, Models, and Applications

David W. Stockburger

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.

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, t_{crit}, 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 t_{crit}= ±
2.228. The sampling distribution model used in a two-tailed t-test is illustrated below:

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 t_{obs} 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 t_{crit} would be positive. For example when a
is set to .05 with ten degrees of freedom (df=10), t_{crit} would be equal to +1.812.

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

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

1. If t_{OBS} = 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 t_{OBS} = -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 t_{OBS} = -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.