lecture13 notes - Stat 312 Lecture 13 Testing on Population Means Moo K Chung [email protected] 3 Testing mean with unknown variance 2 H0 = 0 vs H1 <

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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