Power Calculation
Description: The population mean μ of a parameter x in a population is to be estimated by measuring a sample mean x. It is assumed that the population variance σ2 of x is known. The following two hypotheses about the population mean μ will be tested. Calculate the power of the test as a function of the sample size n and the null cutoff α.
Calculate Power from α and n
| Sample Size | Type I error | Type II error | Power | Test |
|---|---|---|---|---|
| β = | power = 1-β = | one-sided two-sided |
||
| n = | α = | |||
| SE = σ/√n = | Zα = | Zβ = | Effect size D = Dz = |
Assumption and Variable Definition
Assumption
The sample mean x follows a normal distribution with variance σ2/n.
In particular, under H0, x ~ N(μ0, σ2/n);
under HA, x ~ N(μA, σ2/n).
The assumption is valid if x follows a normal distribution. If x does not follow a normal distribution, x can still be approximated by a normal distribution if n is sufficiently large because of the central limit theorem. Note that we assume the population variance σ2 is known. If not, it will have to be estimated from the sample variance and the calculation will be more complicated.
Variable Definition
Standard error SE = σ/√n
Z score: z = (x - μ0)/SE
Effect size D = |μA - μ0|
α: probability of Type I error = area of the shaded red region in the graph.
Zα = Z score of the null cutoff point. For a two-sided test, Zα is the Z score of the null cutoff point closest to μA.
ZA = (μA - μ0)/SE = Z score of μA
Zβ = Zα - ZA = Z score of the null cutoff point measured from z = ZA
β = probability of Type II error = area of the shaded skyblue region in the graph
Dz = |ZA| = scaled effect size