Practice Power of a Test in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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).

Showing a random 20 of 50 problems.

Example 1

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 2

easy
A test rejects a true H0. What kind of error is that, and is power involved?

Example 3

easy
A test has power 0.9. What is the probability of a Type II error?

Example 4

easy
List the four ingredients you must specify before computing power.

Example 5

easy
Increasing the significance level α from 0.05 to 0.10 generally does what to power?

Example 6

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 7

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 8

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 9

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 10

hard
In the same setup as X19, what sample size n is required to achieve power 0.90 at μ=2?

Example 11

easy
If a test has power 0.75, what is β?

Example 12

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 13

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

Example 14

easy
True or false: Power is the probability of a Type I error.

Example 15

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

Example 16

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 17

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

Example 18

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 19

hard
Explain why a study that 'fails to reject H0' is NOT the same as proving H0 true. Use power language.

Example 20

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