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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 setup, the Appendix Table for the standard normal distribution can be used to lookup 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 onesided 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
 WHISTLER
 Economics, Normal Distribution, Variance, Null hypothesis, Statistical hypothesis testing, Type I and type II errors

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