AP®StatisticsCollege BoardRevision NotesUnit 3: Inference for Categorical Data: ProportionsErrors in Hypothesis TestsPower of a Test

Power of a Test (College Board AP® Statistics): Revision Note

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on 3 June 2026

Power of a test

What is the power of a test?

The power of a test is the probability of correctly rejecting the null hypothesis when it was, in reality, false
- A better hypothesis test has a higher power
In practice, you need to be given the actual (true) population parameter to calculate the power
- For example, H₀ assumed $p = \frac{1}{2}$ but actually $p = \frac{1}{3}$
- Power is P(in the critical region, given the actual population parameter is true)

How does power relate to Type II errors?

The power of a test is 1 - P(Type II error)
- Power is the probability of correctly rejecting $H_{0}$ when it is false
- A Type II error means not rejecting $H_{0}$ when it is false
  - These probabilities are complements of each other (so sum to 1)
You ideally want the power of a test to be as high as possible, often 0.8 or higher
- That way it's less likely to produce a Type II error
  - And more likely to reach the correct conclusion

Examiner Tips and Tricks

You should learn the relationship that power is 1 - P(Type II error) as it is not given in the exam.

How do I increase the power of a test?

As the power is 1 - P(Type II error), to increase the power of the test you need to reduce the probability of a Type II error
- This happens when one of the following is changed (and the others are kept the same):
  - The sample size, $n$ , increases
  - The significance level, $α$ , increases
  - The standard error of the hypothesis test decreases
  - The actual (true) population parameter is farther from the null population parameter

Worked Example

An agricultural researcher is testing a new fertilizer to determine if it increases the mean yield of corn per acre compared to the current standard of 140 bushels. The researcher tests $H_{0} : μ = 140$ against $H_{a} : μ > 140$ using a simple random sample of fields and a significance level of $α = 0.05$ . The researcher calculates that if the true mean yield with the new fertilizer is 145 bushels, the power of the hypothesis test is 0.72.

Which of the following statements is true regarding the power of this test?

(A) The power of 0.72 represents the probability that the null hypothesis is actually true, given that the true mean is 145 bushels.

(B) If the researcher were to use a more stringent significance level of $α = 0.01$ instead of 0.05, the power of the test would increase.

(C) If the actual true mean yield of the new fertilizer was 148 bushels instead of 145 bushels, the power of the test would be greater than 0.72.

(D) The probability of making a Type II error in this scenario is calculated as 1−0.05=0.95.

Answer:

The power of a hypothesis test is the probability that the test will correctly reject a false null hypothesis

The probability of a Type II error decreases, and the power increases, when any of the following occur:

the sample size increases
the standard error decreases
the significance level ( $α$ ) increases
or the true parameter value is farther from the null hypothesis

Because 148 bushels is farther away from the null hypothesized value of 140 bushels than 145 bushels is, a true mean of 148 bushels would make it easier to correctly detect the difference, resulting in a power greater than 0.72

Therefore, the correct answer is C

Why the distractors are incorrect:

(A) misinterprets the definition of power
- Power is not the probability that a hypothesis is true, nor is it the probability of an outcome after rejection
- Rather, power assumes a specific alternative reality is true (e.g., the true mean is 145), and represents the probability of successfully rejecting the null hypothesis under those specific circumstances
(B) is incorrect because decreasing the significance level from $α = 0.05$ to $α = 0.01$ makes it harder to reject the null hypothesis, which decreases the power of the test and increases the probability of a Type II error
(D) calculates the error probability incorrectly
- The probability of making a Type I error is defined as the significance level $α$ (0.05)
- The probability of making a Type II error is calculated as 1−power.
- Therefore, the probability of a Type II error in this scenario is 1−0.72=0.28

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