## The Power Function of a Test

When hypothesis testing, thepower of a test is the probability of not committing a Type II error.A Type II error is committed if a false null hypothesis is notrejected. We may think of the power of a hypothesis test as theability of the test to reject a false null hypothesis.

It is quite easy tocalculate the probability of committing a Type I error – rejectingthe null hypothesis when the null hypothesis is true. If the test isconducted at the%level then the probability of rejecting the null hypothesis issincein conducting the hypothesis test we always assume the nullhypothesis is true, and so the probability of committing a Type Ierror is also

If the test is conducted sothat the null hypothesis is rejected if values less than a certainvalueareobserved then the power of the test isandthe power function isForexample, the number of tornadoes to hit a particular townhistorically follows a Poisson distribution with mean,Supposewe now want to asses whether climate change has decreased thefrequency of hurricanes. In the last year there were 3 hurricanes.

The null hypothesis is

The alternative hypothesisis

The power function is

The power of the test ishere

Ifthenthe power is

The power of the test increases in this case if %lambdaincrease. This means the null hypothesis is more likely to berejected ifisfixed at 3 and the probability of a Type II error is reduced. This isnot necessarily a benefit as it means that the probability of a TypeI error is increased. It is in fact impossible to decrease theprobability of a Type I and Type II error simultaneously. Theprobability of each error must be traded so that an optimum isreached.