# Power function

In a parametric test of hypothesis, the power function gives you the probability of rejecting the null hypothesis when the true parameter is equal to . Thus, the graph of a power function is obtained by keeping the null hypothesis fixed and by varying the value of the true parameter.

## Example

Suppose you are testing the null hypothesis that the true parameter is equal to zero:

Suppose that the value of the power function at is

What does this mean? It means that if the true parameter is equal to , then there is a 50% probability that the test will reject the (false) null hypothesis that the parameter is equal to .

## Deriving the power function

In the lecture Hypothesis testing about the mean you can find a detailed derivation of the power function of z-tests and t-tests used to conduct tests of hypothesis about the mean of a normal distribution.

Another example is provided by the lecture Hypothesis testing about the variance, where you can find a derivation of the power function of a Chi-square test used to conduct tests of hypothesis about the variance of a normal distribution.

## More details

You can find a more exhaustive explanation of the concept of power function in the lecture entitled Hypothesis testing.

You can also find other entries in this glossary that are related to hypothesis testing: null hypothesis, alternative hypothesis, Type I error, Type II error.