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 thus, is written as

\[H_0: \mu = \mu_0\text{.}\]

If the observed data \((\mu)\) differs significantly from the supposed population mean \((\mu_0)\), then the the assumption, \(H_0\), is rejected in favor of the alternative hypothesis, \(H_A\). Depending on the specific research question the alternative hypothesis, \(H_A\), is written as

\[H_A: \mu < \mu_0 \quad \text{or} \quad \mu > \mu_0 \quad \text{or} \quad \mu \ne \mu_0 \text{.}\] If however, the data does not provide sufficient evidence to reject the stated hypothesis \((H_0)\), we do not reject \(H_0\) and conclude, that the data does not provide enough evidence to assume that 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, or in other words, the observed variability in the data is due to chance.

In the next sections we show that for the actual calculation of the statistical significance our knowledge on the population parameters is of relevance.