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Unformatted text preview: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection of the null hypothesis. This involves the risk of making an error: x Type I Error the rejection of a true null hypothesis, x Type II Error the failure to reject a null hypothesis when the alternative is true. An approach to hypothesis testing is: choose a significance level . This sets the probability of a Type I error. establish a decision rule. This is determined by the significance level chosen for the test. the probability of a Type II error, , follows. The power of the test is 1 . This is the probability of rejecting the null hypothesis when the alternative hypothesis is true. H 1 H Econ 325 Chapter 9.5 1 How can E , the probability of a Type II error, be determined ? This will be demonstrated for tests of the mean of a normal population when the population variance is known. With this set-up, the Appendix Table for the standard normal distribution can be used to look-up required probabilities. The ideas can be applied to any other hypothesis testing application but computer software must be used to find probabilities. Econ 325 Chapter 9.5 2 The probability of a Type II error can be illustrated with a picture. Consider testing 5 :H against the one-sided alternative 5 :H 1 " Suppose the true population mean is 1.5 . PDF of X with 5 and 1.5 c 5.1 5 if H0 is true if H1 is true significance level probability of a Type II error do not reject o Reject H The probability of a Type II error is the probability that the sample mean is below the critical value c when the true population mean is 5.1 . Econ 325 Chapter 9.5 3 The probability of a Type II error depends on the true value of the population parameter in this case, the population mean. population parameter in this case, the population mean....
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- Spring '10