# Power of a Statistical Test

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The statistical power of the test is the probability of correctly rejecting the null hypothesis when it is false. In other words, it is the probability of not making a type II error in a hypothesis test.

The relationship between the power of the test and the type II error can be expressed as follows:
Power = 1 - β

Where β is the probability of making a type II error or the probability of not rejecting the null hypothesis when it is false.

In other words, the power of the test is the complement of the probability of making a type II error. For example, if the type II error is 0.10, the power of the test would be 1 - 0.10 = 0.90.

It is typically controlled to be greater than 90% by using an appropriate sample size.

For example, if the null hypothesis is that the population mean equals 10. The alternative hypothesis is that the population mean is not equal to 10. The low power of the test would mean that there is a high probability of failing to reject the null hypothesis when the population mean is actually not equal to 10. This would result in a type II error and could lead to incorrect conclusions being drawn.

## Type I, Type II Errors and the Power of Test

A type I error is the probability of rejecting the null hypothesis when it is true. In contrast, a type II error is the probability of not rejecting the null hypothesis when it is false. It is not possible for a researcher to commit both a type I and type II error in a hypothesis test.

However, the risks of type I and type II errors are related. For example, as the alpha value (the probability of making a type I error) increases, the probability of making a type II error (beta) decreases, and the power of the test increases.

To reduce the risks of both type I and type II errors, the researcher can increase the sample size. This will increase the power of the test and reduce the probability of making a type II error. It will also reduce the probability of making a type I error, as the significance level of the test will remain the same.

## Summary

The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false. It is typically controlled to be greater than 90% by using an appropriate sample size. The risks of type I and type II errors are related, and increasing the sample size can reduce both types of errors.

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