One-Tailed Tests
“A ONE-TAILED TEST is applied whenever the hypothesis under consideration specifies a direction” (Meier, Brudney, and Bohte, 2011, p. 198). In a one-tailed test, we are only interested in one tail of the distribution (i.e., values on one side of the mean):
- For a positive relationship, we are interested in the RIGHT TAIL (or UPPER TAIL), i.e., values to the right of/greater than the mean
- For a negative relationship, we are interested in the LEFT TAIL (or LOWER TAIL), i.e., values to the left of/less than the mean
The rejection region is determined by the significance level (α) and the direction of the hypothesized relationship; it usually includes the most extreme 10% (α = 0.10), 5% (α = 0.05), or 1% (α = 0.01) of values in the distribution. Whether these values lie at the bottom (i.e., on the left side) or at the top (i.e., on the right side) of the distribution, and the one-tail test that should be used for hypothesis testing, is determined by the hypothesized relationship:
- A RIGHT-TAILED TEST is used to test for a positive relationship between variables: if the test statistic falls within the top α percent of the distribution, it is in the rejection region, and the null hypothesis is rejected
- Assuming α=0.05 and a 95% confidence interval, our rejection region would be the top 5% of the distribution (i.e., at or above the 95th percentile)
- A LEFT-TAILED TEST is used to test for a negative relationship between variables: if the test statistic falls within the bottom α percent of the distribution, it is in the rejection region, and the null hypothesis is rejected
- Assuming α=0.05 and a 95% confidence interval, our rejection region would be the bottom 5% of the distribution (i.e., at or below the 5th percentile)
Two-Tailed Tests
A TWO-TAILED TEST is used whenever the hypothesis under consideration does not specify a direction; it simultaneously tests for the possibility of both a positive and a negative relationship between variables. Thus, in a two-tailed test, we are interested in both tails of the distribution (i.e., values that fall on both sides of the mean).
The rejection region is determined by the significance level (α), divided equally between the left/lower and right/upper tails; it usually includes:
- for α=0.10, the most extreme 10% of values in the distribution: the bottom 5% of the distribution (i.e., at or below the 5th percentile), and the top 5% of the distribution (i.e., at or above the 95th percentile)
- for α=0.05, the most extreme 5% of values in the distribution: the bottom 2.5% of the distribution (i.e., at or below the 2.5th percentile), and the top 2.5% of the distribution (i.e., at or above the 97.5th percentile)
- for α=0.01, the most extreme 1% of values in the distribution: the bottom 0.5% of the distribution (i.e., at or below the 0.5th percentile), and the top 0.5% of the distribution (i.e., at or above the 99.5th percentile)
If the test statistic falls within the bottom (α/2) percent of the distribution (i.e., the α percentile) or the top (α/2) percent of the distribution (i.e., the 100-α percentile), it is in the rejection region, and the null hypothesis is rejected. Thus, the two-tailed test is more conservative than the one-tailed test because it accounts for the possibility of an effect in either direction.