Measuring the Power of a Test
Suppose a sample of n observations is available from a normal population with mean µ and known variance. Consider testing the null hypothesis:
H0: µ = 5 against the 2-sided alternative
H1: µ 5
The power of any test will depend on:
The above ideas can be demonstrated with an example.
To compute a power function for the test a series of values for the
true population mean is generated. Values are set for the sample size,
the population standard deviation and the significance level.
The power is then calculated for each value of the mean.
The SHAZAM commands (filename:
Figure 1 shows a graph of the power function for a sample size of n=16 and n=100. The population standard deviation is 0.1 and the significance level is 0.05. The figure illustrates that an increase in sample size leads to greater power. The figure also shows that the farther the true mean from the hypothesized value of 5 the greater the power of the test. When the true population mean is 5 the probability that we reject the null hypothesis is 0.05, the significance level of the test.
Figure 2 shows a graph of the power function for significance levels of 0.05 and 0.01. The sample size is n=16 and the population standard deviation is 0.1. The figure illustrates that a smaller significance level gives lower power.
Figure 3 shows a graph of the power function for standard deviations of 0.1 and 0.2. The sample size is n=16 and the significance level is 0.05. The figure illustrates that a larger variance gives lower power.
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