Any hypothesis test starts with the formulation of the null hypothesis and the alternative hypothesis. This section focuses on hypothesis tests for one population mean, $$\mu$$. However, the general procedure applies to any hypothesis test.

The null hypothesis for a hypothesis test concerning a population mean, $$\mu$$, is expressed

$H_0: \mu = \mu_0,$ where $$\mu_0$$ is some number.

The formulation of the alternative hypothesis depends on the purpose of the hypothesis test. There are three ways to formulate an alternative hypothesis (Weiss, 2010):

If the hypothesis test is about deciding, whether a population mean, $$\mu$$, is different from the specified value $$\mu_0$$, the alternative hypothesis is expressed as

$H_A: \mu \ne \mu_0\text{.}$

Such a hypothesis test is called two-sided test.

If the hypothesis test is about deciding, whether a population mean, $$\mu$$, is smaller than the specified value $$\mu_0$$, the alternative hypothesis is expressed as

$H_A: \mu < \mu_0\text{.}$

Such a hypothesis test is called left tailed test.

If the hypothesis test is about deciding, whether a population mean, $$\mu$$, is greater than a specified value $$\mu_0$$, the alternative hypothesis is expressed as

$H_A: \mu > \mu_0\text{.}$

Such a hypothesis test is called right tailed test.

Note that a hypothesis test is called one tailed test if it is either left tailed or right tailed.

$\begin{array}{|l|c|c|} \hline & \text{Two-sided test} & \text{Left-tailed test} & \text{Right-tailed test} \\ \hline \text{Sign in } H_A & \ne & < & > \\ \text{Rejection region } & \text{Both sides} & \text{Left side} & \text{Right side} \\ \hline \end{array}$