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 called one-tailed test 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.
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.