Two-Tailed Test

When the rejection region is taken on both ends of the sampling distribution, the test is called a two-sided test or two-tailed test. When we are using a two-sided test, half of the rejection region equal to \alpha /2 is taken on the right side and the other half equal to \alpha /2 is taken on the left side of the sampling distribution. Suppose the sampling distribution of the statistic is a normal distribution and we have to test the hypothesis {H_o}:\theta = {\theta _o} against the alternative hypothesis {H_1}:\theta \ne {\theta _o} which is two-sided. {H_o} is rejected when the calculated value of Z is greater than {Z_{\alpha /2}} or it is less than  - {Z_{\alpha /2}}.

Thus the critical region is Z > {Z_{\alpha /2}} orZ <- {Z_{\alpha /2}}, and it can also be written as  - {Z_{\alpha /2}} < Z < {Z_{\alpha /2}}.

When {H_o} is rejected, then {Z_1} is accepted. The two-sided test is shown in the figure below.