Two-tailed and one-tailed tests, Applied Statistics

If the test is two-tailed, H1:  μ ≠  μ0 then the test is called two-tailed test and in such a case the critical region lies in both the right and left tails of the sampling distribution of the test statistic, with total area equal to the level of significance as shown in diagram.

If the test is one-tailed either right-tailed or left-tailed), then the test is called a one-tailed test.

For example, to test whether the population mean   μ =  μ0  , we may have the Alternative Hypothesis H1 given by H1:  μ <  μ0 (Left-tailed) or H1:  μ > μ0  (Right-tailed). In this case, the test is a single-tailed or one-tailed test. In the right-tailed test where H1: 

μ >  μ0  , the critical region (or rejection region) z > zα lies entirely in the right tail of the sampling distribution of sample statistic with area equal to the level of significance a.  Similarly, in the left-tailed (H1:  μ < μ0  ), the critical region z < -  zα  lies entirely in the left tail of the sampling distribution of q with area equal to the level of significance α is shown.


2342_one tailed and two tailed test.png

The type of the tests to be applied depends on the nature of the Alternative Hypothesis H1. We apply one-tailed or two-tailed test accordingly as Alternate Hypothesis is one-tailed or two-tailed.

Critical values of z for both two-tailed and one-tailed tests at 10%, 5% and 1% level of significance are given below.

Critical Values of Z

Level of significance (a)




Critical values for Two-tailed tests

   850_one tailed and two tailed test1.png  1.64

   850_one tailed and two tailed test1.png  1.96

   850_one tailed and two tailed test1.png 2.58

Critical values for Left-tailed tests




Critical values for Right-tailed tests




For large samples (n > 30), the sampling distributions of many statistics are approximately normal distribution. In such cases, we can use the results of the table given above to formulate decision rules. We will focus primarily on large samples.

Posted Date: 9/15/2012 3:43:50 AM | Location : United States

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