Understand the differences between one and two-sided hypothesis test.

See the effect of the test statistic on the decision and the PDF of the test statistic.

1. State null and alternative hypothesis

$$ \begin{align} H_0: \theta = \mu_0 \\ H_1: \theta \neq \mu_0 \end{align} $$

2. Decide on the significance level \(\alpha\)

Typically, \(\alpha\) = 5 % is used if not stated otherwise.

3. Calculate the test statistic

$$ \begin{align} t^* = \frac{\overline{\theta}-\mu_0}{\frac{\sigma}{\sqrt{n}}}, \end{align} $$

where \(\mu_0\) is the population mean and \(\theta\) the sample mean.

4. Compute the appropriate p-value based on the alternative hypothesis

In the Illustration section below the plots, the computation of the p-values is shown.

5. Make a decision about the null hypothesis

Compare the p-value with the significance level choosen in step 2. When the p-value is smaller than the significance level \(\alpha\), reject the \(H_0\) hypothesis and conclude \(H_1\). If it is greater than the significance level do not reject \(H_0\).