Power of a Test Examples: 44 Problems with Answers
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check your understanding of Power of a Test.
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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, where β 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 0.80 or above).
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−β.
Common stuck point:The procedure for power of a test is the easy part; the trap is confusing power with α. 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 and β=0.20. Calculate the power and interpret it. If the researcher wants Power=0.90, what must β become?
Answer
Power = 0.80. For Power=0.90, need β=0.10 (achieved by increasing n).
First step
1
Power =1−β=1−0.20=0.80
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Setup·Key insight·Why it works·Common pitfall·Connection
For testing H0:μ=100 vs Ha:μ=105, with σ=10, n=25, α=0.05: calculate the rejection region and power of the test.
Example 3
medium
For testing H0:μ=50 vs Ha:μ>50 with σ=8 and n=64, find the rejection region for Xˉ at α=0.05.
Example 4
medium
A two-sided z-test with α=0.05, σ=10, n=100, true mean shift μa−μ0=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 H0 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. Compute the power of the test.
Example 4
easy
A test has power 0.9. 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 n generally increase or decrease the power of a test?
Example 7
easy
Increasing the significance level α from 0.05 to 0.10 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 H0. Which kind of correct decision is this, and what probability describes how often it happens?
Example 11
medium
A test has α=0.05 and β=0.30. Find the power and the probability of a Type I error.
Example 12
medium
To raise power from 0.6 to 0.8 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 5 units, or detecting a true shift of 2 units (same n, α)?
Example 14
medium
A study is described as 'underpowered.' In terms of β, what does that mean for detecting a real effect?
Example 15
medium
A power analysis gives power =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=50, Design B has n=200. Same α. 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=100 vs Ha:μ>100 has power 0.4 when the true mean is 103. If the true mean were 108 instead (same design), would power be higher or lower?
Example 19
medium
A test has power 0.85. Interpret what the value 0.15 represents here.
Example 20
challenge
A test currently has power 0.5. The researcher considers (i) doubling α, (ii) doubling n, (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 and power 0.80 at a specific alternative. Identify all four probabilities in the decision table: correct retention, Type I error, Type II error, and correct rejection (when H0 is false).
Example 22
challenge
A clinical trial wants power ≥0.9 to detect a 2-point drop, but its budget caps n so power is only 0.6. List two valid adjustments (other than more money for n) and one invalid 'fix.'
Example 23
easy
If a test has power 0.75, what is β?
Example 24
easy
Does a smaller population standard deviation σ tend to increase or decrease power, holding all else fixed?
Example 25
easy
A study has power 0.95. 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 H0. What kind of error is that, and is power involved?
Example 27
medium
Continuing the previous problem, if the true mean is μ=53, compute the power of the test.
Example 28
medium
A clinical trial doubles its sample size from 50 to 200. Holding everything else fixed, what happens to the standard error of Xˉ?
Example 29
medium
A researcher claims power is 0.90 but the planned sample size only delivers power 0.70. What is the realistic Type II error rate?
Example 30
medium
Two studies test the same hypothesis: Study A uses α=0.01 and Study B uses α=0.05. Same n and same effect size. Which has higher power?
Example 31
medium
A power calculation specifies a 'minimum detectable effect' of 5 units. Explain what this number represents.
Example 32
medium
In which scenario is power not meaningful: (a) computing rejection probability under μ=μ0 (the null), (b) under μ=μ0 (an alternative)?
Example 33
medium
A trial has α=0.05, true effect detectable with power 0.5. To achieve power 0.8 (same α, same effect), should n roughly increase, decrease, or stay the same?
Example 34
hard
For a one-sided z-test of H0:μ=0 vs Ha:μ>0 at α=0.05, with σ=4, n=16, what is the power against μ=2?
Example 35
hard
In the same setup as X19, what sample size n is required to achieve power 0.90 at μ=2?
Example 36
hard
Explain why a study that 'fails to reject H0' is NOT the same as proving H0 true. Use power language.
Example 37
hard
Test A has power 0.6 to detect a 3-unit shift. Test B has power 0.6 to detect a 6-unit shift. Same α and σ. Which study has the larger n?
Example 38
hard
A test has power 0.7 at μ=105 for H0:μ=100. Without recomputing, what can you say about its power at μ=110?
Example 39
challenge
In a one-sample z-test, doubling the sample size from n to 2n shifts the standardized effect from δ=Δn/σ to what?
Example 40
challenge
For a two-sided test at α=0.05 and σ known, derive the formula for the minimum n needed to achieve power 1−β against shift Δ.