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.

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

---

Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. 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: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.