The Power Function

THE POWER FUNCTION

The power function depends on the significance level of a test and the sampling distribution of the test statistic under the alternative values of the population parameters. For example, when a Z or t statistic is used to test the hypothesis (H₀) that the population mean μ equals μ₀, the power function equals α at μ₁ = μ₀ and increases as moves away from  μ₀. The power function approaches 1 as μ₁ gets very far from μ₀. Figure 9.1 shows a plot of the power function for a population mean in the simple case when μ0 = 0 and is known, the sample size n = 25, and the population distribution is assumed to be a normal distribution. 

Figure 9.1. Power function for a test that a normal population has mean zero versus a two-sided alterna-tive when the sample size n = 25 and the significance level α = 0.05.

In this case, Z = ( – μ₁)/ ( σ/√n) = ( – μ₁)/( σ/5) = 5( – μ₁)/σ and Z has a standard normal distribution. This distribution depends on μ₁ and σ. We know the value of σ and can take σ = 1, recognizing that although the power depends on μ₁ for the curve in Figure 9.1, to be more general we would replace μ₁ with μ₁/σ for other values of σ. The power is the probability of observing Z in the acceptance region that is P(–C < Z < C), where C is the critical value; consequently, the power depends on the sample size and signif-icance level through C as well as the sample size n through the formula for Z.

Figure 9.2 displays, on the same graph used for n = 25, the comparable results for a sample size n = 100. We see how the power function changes with increased sample size.