Power of a Test Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Power of a Test.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The probability that a hypothesis test correctly rejects a false null hypothesis. Power =P(reject H0∣H0 is false)=1−β= P(\text{reject } H_0 \mid H_0 \text{ is false}) = 1 - \beta, where β\beta is the probability of a Type II error.

Power is your test's ability to detect a real effect when one exists. A test with high power is like a sensitive metal detector—it won't miss a coin buried in the sand. A test with low power is like searching with your eyes—you'll miss things that are actually there. You want power to be high (typically 0.800.80 or above).

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Power is the probability a test correctly rejects a false null: 1−β1-\beta.

Common stuck point: The procedure for power of a test is the easy part; the trap is confusing power with α\alpha. Asking "Am I asking for the probability of correctly rejecting the null GIVEN it is false (the detection rate)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asking for the probability of correctly rejecting the null GIVEN it is false (the detection rate)?

Worked Examples

Example 1

medium
A test has α=0.05\alpha=0.05 and β=0.20\beta=0.20. Calculate the power and interpret it. If the researcher wants Power=0.90, what must β\beta become?

Answer

Power = 0.80. For Power=0.90, need β=0.10\beta=0.10 (achieved by increasing n).

First step

1
Power =1−β=1−0.20=0.80= 1 - \beta = 1 - 0.20 = 0.80

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Example 2

hard
For testing H0:μ=100H_0: \mu=100 vs Ha:μ=105H_a: \mu=105, with σ=10\sigma=10, n=25n=25, α=0.05\alpha=0.05: calculate the rejection region and power of the test.

Example 3

medium
For testing H0:μ=50H_0: \mu = 50 vs Ha:μ>50H_a: \mu > 50 with σ=8\sigma = 8 and n=64n = 64, find the rejection region for Xˉ\bar{X} at α=0.05\alpha = 0.05.

Example 4

medium
A two-sided zz-test with α=0.05\alpha = 0.05, σ=10\sigma = 10, n=100n = 100, true mean shift μa−μ0=2\mu_a - \mu_0 = 2. Approximate the power.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
List four factors that increase the power of a hypothesis test, and explain the direction of each effect.

Example 2

hard
A study fails to reject H0H_0 and concludes 'there is no effect.' Critique this conclusion using the concept of power, and explain what information is needed before accepting this conclusion.

Example 3

easy
The probability of a Type II error is β=0.2\beta = 0.2. Compute the power of the test.

Example 4

easy
A test has power 0.90.9. What is the probability of a Type II error?

Example 5

easy
In words, power is the probability of doing what when the null hypothesis is false?

Example 6

easy
Does increasing the sample size nn generally increase or decrease the power of a test?

Example 7

easy
Increasing the significance level α\alpha from 0.050.05 to 0.100.10 generally does what to power?

Example 8

easy
A larger true effect size (bigger gap between the null and true parameter) does what to power?

Example 9

easy
Can power be computed without specifying a particular alternative (true) parameter value?

Example 10

easy
A test correctly rejects a false H0H_0. Which kind of correct decision is this, and what probability describes how often it happens?

Example 11

medium
A test has α=0.05\alpha = 0.05 and β=0.30\beta = 0.30. Find the power and the probability of a Type I error.

Example 12

medium
To raise power from 0.60.6 to 0.80.8 without increasing the Type I error rate, what should a researcher change?

Example 13

medium
Which test has higher power: detecting a true mean shift of 55 units, or detecting a true shift of 22 units (same nn, α\alpha)?

Example 14

medium
A study is described as 'underpowered.' In terms of β\beta, what does that mean for detecting a real effect?

Example 15

medium
A power analysis gives power =0.95= 0.95 to detect a clinically meaningful difference. Interpret this in plain language.

Example 16

medium
Two designs target the same effect: Design A has n=50n = 50, Design B has n=200n = 200. Same α\alpha. Which has greater power and why?

Example 17

medium
A researcher says 'power is the probability that the null hypothesis is false.' Correct this statement.

Example 18

medium
A test of H0:μ=100H_0: \mu = 100 vs Ha:μ>100H_a: \mu > 100 has power 0.40.4 when the true mean is 103103. If the true mean were 108108 instead (same design), would power be higher or lower?

Example 19

medium
A test has power 0.850.85. Interpret what the value 0.150.15 represents here.

Example 20

challenge
A test currently has power 0.50.5. The researcher considers (i) doubling α\alpha, (ii) doubling nn, (iii) hoping the true effect is larger. Rank which RELIABLY increases power without raising the Type I error rate.

Example 21

challenge
A test has α=0.05\alpha = 0.05 and power 0.800.80 at a specific alternative. Identify all four probabilities in the decision table: correct retention, Type I error, Type II error, and correct rejection (when H0H_0 is false).

Example 22

challenge
A clinical trial wants power ≥0.9\ge 0.9 to detect a 22-point drop, but its budget caps nn so power is only 0.60.6. List two valid adjustments (other than more money for nn) and one invalid 'fix.'

Example 23

easy
If a test has power 0.750.75, what is β\beta?

Example 24

easy
Does a smaller population standard deviation σ\sigma tend to increase or decrease power, holding all else fixed?

Example 25

easy
A study has power 0.950.95. In plain English, what is the chance it will fail to detect a real effect of the specified size?

Example 26

easy
A test rejects a true H0H_0. What kind of error is that, and is power involved?

Example 27

medium
Continuing the previous problem, if the true mean is μ=53\mu = 53, compute the power of the test.

Example 28

medium
A clinical trial doubles its sample size from 5050 to 200200. Holding everything else fixed, what happens to the standard error of Xˉ\bar{X}?

Example 29

medium
A researcher claims power is 0.900.90 but the planned sample size only delivers power 0.700.70. What is the realistic Type II error rate?

Example 30

medium
Two studies test the same hypothesis: Study A uses α=0.01\alpha = 0.01 and Study B uses α=0.05\alpha = 0.05. Same nn and same effect size. Which has higher power?

Example 31

medium
A power calculation specifies a 'minimum detectable effect' of 55 units. Explain what this number represents.

Example 32

medium
In which scenario is power not meaningful: (a) computing rejection probability under μ=μ0\mu = \mu_0 (the null), (b) under μ≠μ0\mu \ne \mu_0 (an alternative)?

Example 33

medium
A trial has α=0.05\alpha=0.05, true effect detectable with power 0.50.5. To achieve power 0.80.8 (same α\alpha, same effect), should nn roughly increase, decrease, or stay the same?

Example 34

hard
For a one-sided zz-test of H0:μ=0H_0: \mu = 0 vs Ha:μ>0H_a: \mu > 0 at α=0.05\alpha = 0.05, with σ=4\sigma = 4, n=16n = 16, what is the power against μ=2\mu = 2?

Example 35

hard
In the same setup as X19, what sample size nn is required to achieve power 0.900.90 at μ=2\mu = 2?

Example 36

hard
Explain why a study that 'fails to reject H0H_0' is NOT the same as proving H0H_0 true. Use power language.

Example 37

hard
Test A has power 0.60.6 to detect a 3-unit shift. Test B has power 0.60.6 to detect a 6-unit shift. Same α\alpha and σ\sigma. Which study has the larger nn?

Example 38

hard
A test has power 0.70.7 at μ=105\mu = 105 for H0:μ=100H_0: \mu = 100. Without recomputing, what can you say about its power at μ=110\mu = 110?

Example 39

challenge
In a one-sample zz-test, doubling the sample size from nn to 2n2n shifts the standardized effect from δ=Δn/σ\delta = \Delta\sqrt{n}/\sigma to what?

Example 40

challenge
For a two-sided test at α=0.05\alpha = 0.05 and σ\sigma known, derive the formula for the minimum nn needed to achieve power 1−β1 - \beta against shift Δ\Delta.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

type i type ii errorshypothesis testingsampling distribution