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(\text{reject } H_0 \mid H_0 \text{ is false}) = 1 - \beta, where \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 0.80 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: Four factors affect power: (1) sample size n—larger is more powerful, (2) significance level \alpha—larger \alpha gives more power but more Type I errors, (3) true effect size—bigger effects are easier to detect, (4) variability—less noise means more power.

Common stuck point: Students confuse power with the p-value. Power is calculated BEFORE the study (planning stage) and depends on the true effect size. The p-value is calculated AFTER data collection.

Worked Examples

Example 1

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

Solution

  1. 1
    Power = 1 - \beta = 1 - 0.20 = 0.80
  2. 2
    Interpretation: if the alternative hypothesis is true, there is an 80% probability of correctly rejecting H_0
  3. 3
    For Power=0.90: \beta = 1 - 0.90 = 0.10; reduce Type II error from 0.20 to 0.10
  4. 4
    Achieving this: increase sample size (most effective way to increase power without changing \alpha)

Answer

Power = 0.80. For Power=0.90, need \beta=0.10 (achieved by increasing n).
Power = P(reject H₀ | H₀ is false) = 1 - β. Higher power means better ability to detect real effects. Increasing sample size is the primary way to increase power while holding α constant. Power depends on: α, effect size, sample size, and population variability.

Example 2

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

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 H_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.
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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