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Unformatted text preview: ∙ For any value of , T Normal n ,1 . ∙ If 0, P T 1.65 .05 because n 0. In other words, the probability of Type I error is its highest when 0; this is why is the least favorable case. 30 ∙ Intuitively, it should be clear that for H : ≤ 0 versus H 1 : 0, rejection rules of the form T c for c 0 are better than other possibilities, such as a T b for real numbers a b . Once we choose a size for a test, we want to maximize power against alternatives. 31 ∙ 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, − 1.65 .05. ∙ If 0, n − 1.65 − 1.65 .05. 32 ∙ For 0, n − 1.65 − 1.65 .05. ∙ As → , → 1. That is, the probability of rejecting H when it is false increases as gets larger. ∙ The next graph plots the power function when n 25. 33 .05 .25 .5 .75 11.5 .5 1 mu Power Function, n = 25, OneSided Alternative, c = 1.65 34 ∙ This graph makes intuitive sense: We should have a better chance of detecting that H 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 ∙ The graph also shows that the probability of rejecting H when it is true is highest at 0. This is why in such problems we often focuses on the simple null H : 0 against the alternative H 1 : 0. 36 ∙ The power function can be used to guide choice of sample size when conducting a survey. For example, suppose that for H : ≤ 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 ∙ To ensure that we reject H : ≤ 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)....
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 Fall '12
 Jeff
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, alternative hypotheses

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