# Power Function of a Test

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When hypothesis testing, the power 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 not rejected. We may think of the power of a hypothesis test as the ability of the test to reject a false null hypothesis.

It is quite easy to calculate the probability of committing a Type I error – rejecting the null hypothesis when the null hypothesis is true. If the test is conducted at the% level then the probability of rejecting the null hypothesis issince in conducting the hypothesis test we always assume the null hypothesis is true, and so the probability of committing a Type I error is also

If the test is conducted so that the null hypothesis is rejected if values less than a certain valueare observed then the power of the test isand the power function isFor example, the number of tornadoes to hit a particular town historically follows a Poisson distribution with mean,Suppose we now want to asses whether climate change has decreased the frequency of hurricanes. In the last year there were 3 hurricanes.

The null hypothesis is

The alternative hypothesis is

The power function is

The power of the test is here

Ifthen the power is

The power of the test increases in this case if %lambda increase. This means the null hypothesis is more likely to be rejected ifis fixed at 3 and the probability of a Type II error is reduced. This is not necessarily a benefit as it means that the probability of a Type I error is increased. It is in fact impossible to decrease the probability of a Type I and Type II error simultaneously. The probability of each error must be traded so that an optimum is reached.