Power of a Test Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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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?

Solution

  1. 1
    Power =1−β=1−0.20=0.80= 1 - \beta = 1 - 0.20 = 0.80
  2. 2
    Interpretation: if the alternative hypothesis is true, there is an 80% probability of correctly rejecting H0H_0
  3. 3
    For Power=0.90: β=1−0.90=0.10\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 α\alpha)

Answer

Power = 0.80. For Power=0.90, need β=0.10\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.

About Power of a Test

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.

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More Power of a Test Examples

Example 2 hard

For testing [formula] vs [formula], with [formula], [formula], [formula]: calculate the rejection re

Example 3 easy

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

Example 4 hard

A study fails to reject [formula] and concludes 'there is no effect.' Critique this conclusion using

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