e-Statistics

Power of Test

What is the probability that we incorrectly reject the null hypothesis $ H_0$ when it is actually true? The probability of such an error is called the probability of type I error, and is exactly the significance level $ \alpha =$ . But how about the probability that we incorrectly accept the null hypothesis $ H_0$ when it is actually false? Such probability $ \beta$ is called the probability of type II error.

Given the current estimate $ \mu =$ of population mean and $ \sigma =$ of standard deviation, it is possible to find

$ (1-\beta) =$
The value $ (1-\beta)$ is known as the power of the test, indicating how correctly $ H_A$ can be accepted when it is actually true in the following hypothesis test problem

$ H_A:\hspace{0.05in}\mu$ $ \mu_0 =$

The power $ (1-\beta)$ of the test can be calculated with a specific choice of the sample size

$ n =$

The power of the test increases as the sample size $ n$ increases. Therefore, we can achieve the desired power $ (1-\beta)$ of the test instead by increasing a sample size $ n$. Furthermore, having prescribed the power $ (1-\beta)$ and the sample size $ n$, it is possible to derive the corresponding value $ \mu_0$.