Section 5.5

Fundamentals of Social Statistics by Adam J. McKee

One-tailed vs. Two-tailed Tests

Recall that accepting or rejecting the null hypothesis is based on where a particular test statistic falls within an associated probability distribution.  That is, we reject the null hypothesis if the test statistic is sufficiently improbable.  We determine this by observing if the probability of such a result falls within the extreme tails of the probability curve.  Tails are defined by alpha (α).

It must be considered that if we set alpha at .05 (5%), we are by default saying that we will reject the null hypothesis if the test statistic falls within the most extreme 2.5% found at both ends of the distribution.  In other words, we are considering two tails of the distribution, not just one.  To put this in terms of standard deviation units, we would reject if the test statistic falls past 1.96 standard deviation units to the left or if it falls past 1.96 standard deviation units to the right of the probability distribution.  Since we are considering both tails, this type of test is known as a two-tailed test.

If we are confident for theoretical reasons that the difference between means is in a particular direction, we can specify a one-tailed test.  This moves our 5% rejection region into one tail of the probability distribution.  This increases power so long as the mean difference we observe in our sample data is in the hypothesized direction.  In our example of a two-tailed test, we reject the null hypothesis if the test statistic is larger than 1.96.  For the one tailed-test, we reject the null hypothesis if the tests statistic is larger than 1.65.  This lower standard makes it much easier for a researcher to reject a null hypothesis.

Since it is more powerful, it may seem like a good idea to use a one-tailed test all of the time.  The problem is that we have to follow some rules that make this less attractive.  The most important rule is that if we use a one-tailed test, we must specify that a priori.  That is, we have to specify the direction of the mean differences before we conduct the analysis.  If we later observe that the means were different in the opposite direction than we hypothesized, we must conclude that our findings are not significant.

Key Terms

Hypothesis, Sample, Population, Generalization, Inference, Test Statistic,  Research Hypothesis, Null Hypotheses, p-values, Alpha Level, Type I Error, Type II Error, Power, Assumptions, One-tailed Test, Two-tailed Test

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Last Modified:  06/03/2021

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