Power Calculations for One-Sample Z Test
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Consider the case where the null hypothesis is that the population mean, μ0, is 100 with a population standard deviation, σ, of 10. If the true population, μ1, mean is 105 and we use a sample of size n=25, how likely is it that we will be able to reject the null hypothesis (one-tailed alpha = .05)? That is, how much statistical power do we have?
The WISE Power Applet shown below indicates that the power is .804. The calculations below show how we could compute the power ourselves.
Derivation of a formula for power computation
If we observe a sample mean greater than the critical value indicated by the red dashed line, we will reject the null hypothesis. Let us refer to this critical value as C.
The critical value C is defined on the blue null distribution as the value that cuts off the upper .05 (i.e., alpha) of the blue distribution. We can find the Z score for this value by using a standard Z table or theWISE p–z converter applet. Let us call this score Zα because it is defined on the null distribution by alpha. In our example, Zα = 1.645. Thus, the critical value C is 1.645 standard errors greater than the mean of the null distribution, μ0.
In our example,
Similarly, we can generate a formula for the value of C on the red sampling distribution for the alternate hypothesis, μ1 = 105. Beta error corresponds to the portion of the red curve that falls below C, while statistical power corresponds to the portion of the red curve that falls above C.
Let us use Zβ as the label for the standardized score on the red distribution corresponding to C because the beta error is defined as the portion of the red curve that falls below C.
, where Zβ is a negative number in our example.
We can set the two equations for C equal to each other, giving
Rearranging terms gives
where d is a common measure of effect size: d = (μ1 – μ0) / σ.
The result is this elegant formula:
This formula expresses the relationship between four concepts. If we know any three, we can compute the fourth. An easy way to remember these four concepts is with the mnemonic BEAN:
- B = beta error rate, represented by Zβ. Power is (1 – beta error rate).
- E = effect size, represented by d
- A = alpha error rate, represented by Zα
- N = sample size, represented by n