Thegraphbelowshowsthepowerfunctionforthetwotailedtest

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Unformatted text preview: for the probability can be found by following the calculation steps shown in part (a). 8 Chapter 10.5 (c) Suppose the true mean is μ = 31 . With a sample size of n = 36, what is the probability of a Type II error, using this decision rule ? β = P( X > 30 .8 μ = 31 ) This is the probability that the sample mean is not in the rejection region when the true mean is 31. This is found as: β ⎛ X − 31 30 .8 − 31 ⎞ ⎟ = P⎜ ⎜ σ n > 3 36 ⎟ ⎠ ⎝ = P( Z > − 0 . 4 ) = P( Z < 0 . 4 ) = 0.6554 look ‐ up in Appendix Table 1 For a population mean of 31, the power of the test is: 1 − β = 1 − 0.6554 = 0.3446 This probability can be calculated for any value of μ (the true mean). The value of β and the power will be different for different values of μ . 9 Chapter 10.5 Now consider testing the null hypothesis: H0 : μ = 5 against the two‐sided alternative: H1 : μ ≠ 5 For a given decision rule, the probability of a Type II error can be calculated for different values of the true population mean. The graph below shows the power function for the two‐tailed test with a sample size of n = 16 and n = 100. Power Function σ = 0.1 , α = 0.05 1 power n = 100 n = 16 .5 0.05 4.95 5 mean 5.05 The figure illustrates that an increase in sample size leads to greate...
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