Practice Power of a Test in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick 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).

Showing a random 20 of 50 problems.

Example 1

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 2

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

Example 3

easy
A test has power 0.90.9. 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\alpha from 0.050.05 to 0.100.10 generally does what to power?

Example 6

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 7

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 8

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 9

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 10

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 11

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

Example 12

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 13

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 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 5050 to 200200. Holding everything else fixed, what happens to the standard error of Xˉ\bar{X}?

Example 17

medium
A power analysis gives power =0.95= 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\alpha = 0.01 and Study B uses α=0.05\alpha = 0.05. Same nn and same effect size. Which has higher power?

Example 19

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

Example 20

easy
In words, power is the probability of doing what when the null hypothesis is false?
← Back to Power of a Test Worked Examples Formula Explained