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Power function

In a parametric test of hypothesis, the power function [eq1] gives you the probability of rejecting the null hypothesis when the true parameter is equal to $\theta $. 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: [eq2]

Suppose that the value of the power function at $\theta =1$ is [eq3]

What does this mean? It means that if the true parameter is equal to 1, then there is a 50% probability that the test will reject the (false) null hypothesis that the parameter is equal to 0.

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.

Keep reading the glossary

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Next entry: Precision matrix

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