One-Tailed Test

When the alternative hypothesis {H_1} is one-sided like \theta > {\theta _o} or \theta < {\theta _o}, then the rejection region is taken only on one side of the sampling distribution. This is called a one-tailed test or one-sided test. When {H_1} is one-sided to the right like \theta > {\theta _o}, the entire rejection region equal to \alpha is taken in the right end of the sampling distribution.

Here the test is called one-sided to the right. The hypothesis {H_o} is rejected if the calculated value of a statistic, say Z, falls in the rejection region. The critical value is {Z_\alpha } which has the area equal to \alpha to its right. The rejection region and acceptance region are shown in the figure below. The null hypothesis {H_o} is rejected when Z (calculated)  > {Z_\alpha }.


one-tail-right

If the alternative hypothesis is one-sided to the left like \theta < {\theta _o}, the entire rejection region equal to \alpha is taken on the left tail of the sampling distribution. The test is called one-sided or one-tailed to the left. The critical value is  - {Z_\alpha }, which cuts off the area equal to \alpha to its left. The critical region is Z < - {Z_\alpha } and is shown in the figure below.


one-tail-left

For some important values of \alpha , the critical values of Z for two-tailed and one tailed tests are given below:

Critical Value of Z

\alpha
Two-sided test
One-sided test
One-sided to the left
0.10{\text{ }}(10\% )
 - 1.645and  + 1.645
 + 1.282
 - 1.282
0.05{\text{ }}(5\% )
 - 1.96and  + 1.96
 + 1.645
 - 1.645
0.02{\text{ }}(2\% )
 - 2.326and  + 2.326
 + 2.054
 - 2.054
0.01{\text{ }}(1\% )
 - 2.575and  + 2.575
 + 2.326
 - 2.326