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

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Stat 312: Lecture 14 Large sample test Moo K. Chung [email protected] October 28, 2004 1. Testing mean μ with unknown variance σ 2 . H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: t = ¯ x - μ 0 s/ n . Rejection region for level α test: z ≤ - t α,n - 1 . Ex. IQ of a dog, X i N ( μ,σ 2 ) , where σ is unknown. Test H 0 : μ = 100 vs. H 1 : μ < 100 at level α = 0 . 05. > x<-c(30, 25, 70, 110, 40, 80, 50, 60, 100, 60) > t=(mean(x)-100)/(sd(x)/sqrt(10)) > t [1] -4.205156 > qt(0.05,9) [1] -1.833113 A simpler method is to use command t.test . >help(t.test) ... t.test(x,alternative=c("two.sided", "less", "greater"),conf.level = 0.95) ... > t.test(x,mu=100,alternative="less", conf.level=0.95) One Sample t-test data: x t = -4.2036, df = 9, p-value = 0.001147 alternative hypothesis: true mean is less than 100 95 percent confidence interval: -Inf 78.8531 sample estimates: mean of x 62.5 2. Testing on mean μ when the sample size is large. H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: z = ¯ x - μ 0 s/ n . Rejection region for level α test: z ≤ - z α . 3. Let p be the proportion of a population with a specified property. Assume the sample size to
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Unformatted text preview: be large ( n > 40). Large-sample level. α test for H : p = p vs. H 1 : p < p test statistic: z = ˆ p-p √ p (1-p ) /n . rejection region: z <-z α . Example 8.11. 47 out of 102 doctors did not know the generic name for the drug methadone.What is methadone? Methadone is a rigorously well-tested medication that is safe and efficacious for the treatment of narcotic withdrawal and dependence. Can you conclude that fewer than half of all doctors know the generic name for methadone? Test it at 1% level. > p<-47/102 > (p-0.5)/sqrt(0.5*0.5/102) [1] -0.792118 > qnorm(0.01) [1] -2.326348 4. Small sample size. Example 8.13. > pbinom(15,20,0.9) [1] 0.04317450 > pbinom(16,20,0.9) [1] 0.1329533 Homework V. Due Nov 11. 9:30pm. Exercise 8.18, 8.24, 8.26, 8.32, 8.38, 8.40. Second midterm (Nov 18) will cover Homework III. IV. and V. ma-terials....
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