 The Power of the Test

P (Type II Error) = β

P (Type I Error) = level of significance = α

The consequence of a small α is large β.

Be careful, (1-β) is not α

because (1-β) = the power of the test.

The probability of rejecting the null hypothesis when it is false is called the power of the test or simply 1 - β. If the null hypothesis is false, then it should be rejected, so, the power of the test is actually giving us the probability of rejecting the null hypothesis when it should be rejected.

Therefore, the higher the power of the test, the higher the probability of rejecting the null hypothesis when it is wrong, and the more likely and correctly the test will reject the null hypothesis without making any type of error.

To find out the power of the test, let’s see the same example,

suppose we have hypotheses such as

Ho: μ0 = 500

(null hypothesis with specific population mean)

Ha: μ > 500

(alternative hypothesis with an assumption that

the population mean could be greater than μ0 )

for a sample size of n = 40

with population standard deviation (σ) of 115

at the level of significance α that is probability of making type I error is 0.01

Find the power of the test.

So, first get the P (Type II Error)

P (Type II Error) = 0.8389

thus,                β = 0.8389

Power of the Test = 1 - β = 1 - 0.8189 = 0.1611

Therefore, the power of the test which is actually the probability of rejecting the null hypothesis when it is wrong is 0.1611. The power of the test is a small value because the probability of making type II error that is not rejecting the null hypothesis when it should be rejected because it is wrong is large.