Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 12 Errors in hypothesis testing Moo K. Chung [email protected] Oct 20, 2004 Type I error : you reject H 0 when H 0 is true. P ( Type I error ) = P ( reject H 0 | H 0 true ) = α. The resulting α is called the significance level of the test and the corresponding test is called a level α test . We will use test procedures that give α less than a specified level (0.05 or 0.01 usually). I believe that dogs are as smart as people. Assume IQ of a dog follows X i N ( μ, 10 2 ) . IQ of 10 dogs are measured: 30, 25, 70, 110, 40, 80, 50, 60, 100, 60. We want to test if dogs are as smart as people by testing H 0 : μ = 100 vs. H 1 : μ < 100 . (1) Let ¯ X be a test statistic and R = ( -∞ , 90) to be a rejection region. Let’s compute the probabil- ity of making Type I error based on this testing procedure. Under the assumption H 0 is true, X i N (100 , 10 2 ) . Under this condition, ¯ X N (100 , 10) and α = P ( ¯ X < 90) . >help(pnorm) ... Usage: ... pnorm(q, mean=0, sd=1,...) ... > pnorm(90,100,sqrt(10)) [1] 0.0007827011 By using this test procedure, it is highly unlikely to make Type I error. (2) Let’s see what happens when we change the test precedure. When R = ( -∞ , 95) with the same test statistic ¯ X , α = P ( ¯ X < 95) . > pnorm(95,100,sqrt(10)) [1] 0.05692315 70 90 110 0.0 0.4 0.8 a pnorm(a, 100, sqrt(10)) 70 90 110 0.0 0.4 0.8
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