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. It is called 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.

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 Figure. 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 Figure.

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


one-tail-left

Critical Value of Z

\alpha

Two-sides 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

 

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