Econ 325 chapter 95 4 the probability of a type ii

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Econ 325 – Chapter 9.5 4 The probability of a Type II error depends on the true value of the population parameter – in this case, the population mean. In practice, the true value of the population mean is unknown. From inspection of the above graph, some general results can be stated. By keeping the sample size and population variance the same then: the probabilities of a Type I error and Type II error are inversely related. Lowering the significance level of a test (the probability of a Type I error) increases the chance of a Type II error. the closer the true value of the population mean μ to the value stated in the null hypothesis, the greater the probability of a Type II error and the lower the power ( β - 1 ) of the test. That is, it is more difficult to detect differences between the null and alternative hypotheses. For example, in the above graph, the probability density function for the sample mean when the true population mean is 05 . 5 = μ is shifted to the left compared to the PDF when the mean is 1 . 5 = μ . This increases the the probability of a Type II error.
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Econ 325 – Chapter 9.5 5 Example Assume the population standard deviation is: 3 = σ . From a sample of n = 9 the calculated sample mean is: 2 . 48 x = . (a) At a 10% significance level, test 50 : H 0 μ against 50 : H 1 < μ The test statistic is: 1.8 9 3 50 48.2 n 50 - = - = σ - = x z The 10% critical value is 1.282 = c z (since the population variance is viewed as known, the critical value is found in the t-distribution Appendix Table in the final row for degrees of freedom ). The results show c z z - < to give evidence to reject the null hypothesis. Econ 325 – Chapter 9.5 6 (b) Find the power of a 10% level test when the true mean is 49 . The null hypothesis is rejected for: 1.282 9 3 50 n 50 - < - = σ - x x or, by rearranging, the null hypothesis is rejected for: 48.718 1.282 50 = - < x The probability of a Type II error is the probability that the sample mean is not in the rejection region when the true mean is 49 . This is stated as: ) 49 718 . 48 X ( P = μ > = β This is found as: Table Normal Standard the in up - look 61 . 0
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