# 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) $> {Z_\alpha }$.

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.

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 $0.10{\text{ }}(10\% )$ $– 1.645$and $+ 1.645$ $+ 1.282$ $– 1.282$ $0.05{\text{ }}(5\% )$ $– 1.96$and $+ 1.96$ $+ 1.645$ $– 1.645$ $0.02{\text{ }}(2\% )$ $– 2.326$and $+ 2.326$ $+ 2.054$ $– 2.054$ $0.01{\text{ }}(1\% )$ $– 2.575$and $+ 2.575$ $+ 2.326$ $– 2.326$