Power of a test
A decision rule about whether to accept or reject H0 can result in one of two types of error. The probabilities of making these errors describe the risks involved in the decision.
Instead of the probability of a Type II error, it is common to use the power of the test, defined as one minus the probability of a Type II error,
The power of a test is the probability of correctly rejecting H0 when it is false.
When the alternative hypothesis includes a range of possible parameter values (e.g. µ ≠ 0), the power depends on the actual parameter value.
Decision | |||
---|---|---|---|
accept H0 | reject H0 | ||
Truth | H0 is true | Significance level = P (Type I error) |
|
HA (H0 is false) | P (Type II error) | Power = 1 - P (Type II error) |
Increasing the power of a test
It is clearly desirable to use a test whose power is as close to 1.0 as possible. There are three different ways to increase the power.
In CAST, we only describe the most powerful type of decision rule to test any hypotheses, so you will not be able to increase the power by changing the decision rule.
When the significance level is fixed, increasing the sample size is therefore usually the only way to improve the power.
Illustration
The following diagram again investigates decision rules for testing the hypotheses
H0 : μ = 10
HA : μ > 10
based on a samples from a normal population with known standard deviation σ = 4. We will fix the significance level of the test at 5%.
The top half of the diagram shows the normal distribution of the mean for a sample of size n = 16. Use the slider to increase the sample size and observe that: