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.

**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 (*H*_{0})
that the population mean μ equals μ_{0}, the power function equals α at μ_{1} = μ_{0} and increases as moves away from μ_{0}. The power function approaches 1 as μ_{1} gets very far from μ_{0}. Figure 9.1 shows a plot of the power function for a population mean in
the *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* = ( – μ_{1})/ ( σ/√n) = ( – μ_{1})/( σ/5) = 5( – μ_{1})/σ and *Z* has a standard normal
distribution. This distribution depends on μ_{1} and σ. We know the value of σ and can take σ = 1, recognizing that although the power depends on μ_{1} for the curve in Figure 9.1, to be more general we
would replace μ_{1} with μ_{1}/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.

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