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

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Stat 312: Lecture 13 Testing on Population Means Moo K. Chung [email protected] October 26, 2004 1. Since we can minimize both type I and type II errors simultaneously in constructing a test procedure, we will control for fixed type I error (level α ) and make type II error as small as possible. The corresponding test procedure is called a level α test . 2. Testing mean μ with known variance σ 2 . H 0 : μ = μ 0 vs. H 1 : μ < μ 0 Test statistic: z = ¯ x - μ 0 σ/ n , which is a pivot under H 0 . A pivot is a statistic whose distribution is independent of population paramters. Rejec- tion region for level α test: z ≤ - z α . Ex. Intelligence quotient (IQ) is a number used to express the relative intelligence of a person. An average person has the IQ of 100. Assume IQ of a dog follows X i N ( μ, 10 2 ). The IQs 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 at level α = 0 . 05. > x<-c(30, 25, 70, 110, 40, 80, 50, 60, 100, 60) > mean(x) [1] 62.5 > z<-(mean(x)-100)/(10/sqrt(10)) > z [1] -11.85854 > qnorm(0.05)
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Unformatted text preview: [1] -1.644854 > qnorm(0.025) [1] -1.959964 Since z <-z . 05 =-1 . 64, we reject H at α = 0 . 05 level. 3. Testing mean μ with unknown variance σ 2 . H : μ = μ vs. H 1 : μ < μ Test statistic: t = ¯ x-μ 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 : μ = 100 vs. H 1 : μ < 100 at level α = 0 . 05. > 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 Review problems Example 8.6., 8.7., 8.8., 8.9....
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