In the previous example, you tested a research hypothesis that predicted not only that the sample mean would be different from the population mean but that it would be different in a specific direction—it would be lower. This test is called a directional or one‐tailed test because the region of rejection is entirely within one tail of the distribution.
Some hypotheses predict only that one value will be different from another, without additionally predicting which will be higher. The test of such a hypothesis is nondirectional or two‐tailed because an extreme test statistic in either tail of the distribution (positive or negative) will lead to the rejection of the null hypothesis of no difference.
Suppose that you suspect that a particular class's performance on a proficiency test is not representative of those people who have taken the test. The national mean score on the test is 74.
The research hypothesis is:
The mean score of the class on the test is not 74.
Or in notation: H a : μ ≠ 74
The null hypothesis is:
The mean score of the class on the test is 74.
In notation: H 0: μ = 74
As in the last example, you decide to use a 5 percent probability level for the test. Both tests have a region of rejection, then, of 5 percent, or 0.05. In this example, however, the rejection region must be split between both tails of the distribution—0.025 in the upper tail and 0.025 in the lower tail—because your hypothesis specifies only a difference, not a direction, as shown in Figure 1(a). You will reject the null hypotheses of no difference if the class sample mean is either much higher or much lower than the population mean of 74. In the previous example, only a sample mean much lower than the population mean would have led to the rejection of the null hypothesis.
Figure 1.Comparison of (a) a two‐tailed test and (b) a one‐tailed test, at the same probability level (95 percent).