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 is
sincein 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 is
andthe power function is
Forexample, 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
If
thenthe 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.