The simplest form of a hypothesis test is the test for one population mean. In this test, we compare the mean, $\mu$, obtained from sampling (observations) against a supposed population mean, $\mu_{0}$. The null hypothesis states that the mean and the population mean are equal and is thus written as:
$$H_{0}\ :\ \mu = \mu_{0}$$If the mean of the observed data ($\mu$) differs significantly from the supposed population mean ($\mu_{0}$), then the assumption, $H_{0}$, is rejected in favour of the alternative hypothesis, $H_{A}$. Depending on the specific research question the alternative hypothesis, $H_{0}$, is written as:
$$H_{A}\ :\ \mu < \mu_{0}\ \ \ or\ \ \ \mu > \mu_{0}\ \ \ or\ \ \ \mu \ne \mu_{0}$$If, however, the data does not provide sufficient evidence to reject the stated hypothesis ($H_{0}$), we do not reject $H_{0}$. In this case, we conclude that the data does not provide enough evidence to assume the observed mean ($\mu$) is different from the supposed population mean ($\mu_{0}$). The observed variability in the data is attributed to the inherently probabilistic nature of the data generation process. In other words, the observed variability in the data is due to chance.
In the following sections, we show that our knowledge of the population parameters is of relevance for the actual calculation of the statistical significance.
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.