Any hypothesis test starts with formulating 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: A hypothesis test is calledone-tailedtest if it is either left-tailed or right-tailed.

Two-sided test | Left-tailed test | Right-tailed test | |
---|---|---|---|

Sign in $H_{A}$ | $\ne$ | $<$ | $>$ |

Rejection region | Both sides | Left side | Right side |

**Citation**

The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: *Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis
using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.*