A one-tailed test (one-sided test) is a statistical test that considers a change in only one direction. In such a test, the alternative hypothesis has either a < (less than sign) or > (greater than sign) i.e. we consider either an increase or reduction but not both.

**Breaking Down a One-tailed Test**

A one-tailed test directs all of the significance level (α) to the testing of statistical significance in one direction. In other words, we aim at testing the possibility of a change in one direction and completely disregard the possibility of a change in the other direction.

Suppose our significance level is 5%: This means we allot 0.05 of the total area in one tail of the distribution of our test statistic. Let’s look at an example:

Let’s assume that we are using the standardized normal distribution to test the hypothesis that the population mean is equal to a given value X using the data from a sample drawn from the population of interest. Our null hypothesis can be expressed as:

H_{o}: μ = X

If our test is one-tailed, the alternative hypothesis will test if the mean is significantly greater than X or significantly less than X but NOT both.

**Case1 (α = 0.05)**

H_{1}: μ < X

The mean is significantly less than X if the test statistic is in the bottom 5% of the probability distribution. This bottom area is known as the critical region (rejection region). i.e. we would reject the null hypothesis if the test statistic is less than -1.645.

**Case 2**

H_{1}:μ > X

We would reject the null hypothesis only if the test statistic is greater than the upper 5% point of the distribution. i.e. reject H_{0} if the test statistic is greater than 1.645.

**A Two-tailed Test**

A two-tailed test considers the possibility of a change in either direction. It looks for a statistical relationship in both the positive and the negative directions of the distribution. Thus, it allows half the value of ? to the testing of statistical significance in one direction and the other half to the testing of the same in the opposite direction. A two-tailed test may have the following set of hypothesis:

H_{1}: μ = X

H_{1}: μ *≠* X

Using our earlier example, if we were to carry out a two-tailed test, we would reject H_{0 }if the test statistic would be either less than the lower 2.5% point or greater than the upper 2.5% point of the normal distribution.

*Reading 12 LOS 12b:*

*Distinguish between one-tailed and two-tailed tests of hypotheses.*