Let H0 be the null hypothesis, Ha the alternate hypothesis, mu the population mean and mu-H0 the hypothesized mean. A researcher sets H0: mu = mu-H0 and Ha: mu < mu-H0. In this situation, the researcher would apply:

With Ha stated as mu less than the hypothesized mean, the rejection region lies wholly in the left tail, giving a left-tailed test.

A one-directional alternative pointing below the hypothesized value defines a lower or left-tailed test.

A right-tailed test applies when the alternative claims mu is greater than the hypothesized mean.

A two-tailed test applies when the alternative is simply mu not equal to the hypothesized mean.

Half-tailed test is not a recognised category in hypothesis testing.

The direction of the inequality in Ha alone fixes which tail carries the critical region.