In fact this convergence holds if we replace 165 by

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(In fact, this convergence holds if we replace 1.65 by any critical value c .) 24
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We have shown that the asymptotic power function for this example is 0, 0 0 .05 1, 0 Unfortunately, this is unhelpful for determining what the likely power is for a given sample size. Most sensible tests for this problem will have exactly the same asymptotic power function. In fact, if one uses only the even-numbers observations (so only half of the data), the asymptotic power function is the same. 25
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In the case of a normal population, we computed the exact power function for any n , and it does not look much like the asymptotic power function (although we can see it is converging to the asymptotic power function). 26
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0 .2 .5 1 -1 -.5 0 .5 1 mu exact power asymptotic power Exact Power (n = 25) and Asymptotic Power 27
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.05 .2 .5 1 -1 -.5 0 .5 1 mu exact power asymptotic power Exact Power (n = 100) and Asymptotic Power 28
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