Power and sample size

Type II error

The diagram shows the distribution of the test statistic when the null hypothesis is false (the right hand, pink curve) alongside the distribution when the null hypothesis is true (the left hand, blue curve).

When we conduct a hypothesis test we make our decision to reject or not reject the null hypothesis based on the value of the t-statistic under the assumption that the null hypothesis is true.

But we will fail to reject a false null hypothesis (i.e. make a type II error) if the value of the t-statistic falls on the left hand side of the cut-off value of the blue curve - so the pink shaded area indicates the probability of making a type II error ().

If the value of the t-statistic falls on the right hand side of the cutoff value of the blue curve we will correctly reject a false null hypothesis.

The area under the pink curve to the right of the cutoff value shows the probability of correctly rejecting a false null hypothesis and is equal to 1 - . This is also called the power of the study. Good studies have at least 90% power to correctly reject a null hypothesis that is false.