31 here we can easily compute the power function p t

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31
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Here we can easily compute the power function: P T 1.65| P n X ̄ n 1.65 n | P n X ̄ 1.65 n | 1 1.65 n n 1.65 using the symmetry of the standard normal distribution. When 0, 0 1.65 .05. If 0, n 1.65 1.65 .05. 32
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For 0, n 1.65 1.65 .05. As , 1. That is, the probability of rejecting H 0 when it is false increases as gets larger. The next graph plots the power function when n 25. 33
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.05 .25 .5 .75 1 -1 -.5 0 .5 1 mu Power Function, n = 25, One-Sided Alternative, c = 1.65 34
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This graph makes intuitive sense: We should have a better chance of detecting that H 0 is false when the true value is farther from the null. At 1, the power function is virtually one: 1 25 1 1.65 3.35 .9996 The power function is for a fixed value of n ( n 25 in the graph). But it is clear from n 1.65 that the power increases as n for any 0. (More later on asymptotic properties of tests.) 35
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The graph also shows that the probability of rejecting H 0 when it is true is highest at 0. This is why in such problems we often focuses on the simple null H 0 : 0 against the alternative H 1 : 0. 36
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The power function can be used to guide choice of sample size when conducting a survey. For example, suppose that for H 0 : 0, we would like the power to be .80 against the alternative value 1 .25. Then we need to find n such that .25 n 1.65 .80 so .25 n 1 .80 1.65 n 4 1 .80 6.6 2 99.3 37
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To ensure that we reject H 0 : 0 in at least 80% of random samples when the true population mean is 1, we need n to be at least 100 (because we cannot have 99.3 observations). I used the invnormal function in Stata to compute 1 .80 . For the entire expression, we can use di : . di (4*invnormal(.80) 6.6)^2 99.330822 38
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EXAMPLE : Suppose we are sampling from a count distribution – and we do not specify a particular class of distributions. Let 0 be the mean and 2 0 the variance. Consider the null and alternative: H 0 : 2 H 1 : 2 (where the null holds for the Poisson distribution but others, too). The null is composite because it can hold for any positive value of . Of course the alternative is composite, too. 39
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