Stat 312: Lecture 6 - Confidence Intervals II, Study notes of Mathematical Statistics

Download Stat 312: Lecture 6 - Confidence Intervals II and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 312: Lecture 6 Confidence Intervals II. Moo K. Chung mchung@stat.wisc.edu February 6, 2003 Concepts 1. Let Xi ∼ N(µ, σ2) with known σ2 and unknown µ. 100(1 − α)% confidence interval for µ is. µ̂L = x̄ − zα/2 · σ/ √ n, µ̂U = x̄ + zα/2 · σ/ √ n. 2. The sample size is inversely related to the width of confidence interval. 3. Central Limit Theorem. Let X1, · · · , Xn be a ran- dom sample with mean µ and variance σ2. For large n, Z = X̄ − µ σ/ √ n ∼ N(0, 1). 4. If n is sufficiently large, approximate 100(1−α)% confidence interval for µ is x̄ ± zα/2s/ √ n, where s is the sample standard deviation. In-class problems Continuing Exercise 6.25, construct 98% confidence in- terval. Example 7.4. Response time ∼ N(µ, σ2), σ = 25. Find the sample size n that ensures 95% CI with a width of 10. Example 7.6. Alternating current (AC) voltage data > data(xmp07.06) > attach(xmp07.06) > str(xmp07.06) ‘data.frame’:48 obs. of 1 variable: $ C1: int 62 50 53 57 ... > boxplot(C1) > mean(C1) [1] 54.70833 > sd(C1) 40 45 50 55 60 65 [1] 5.230672 >mean(C1)-qnorm(0.975)*sd(C1)/sqrt(length(C1)) [1] 53.2286 >mean(C1)+qnorm(0.975)*sd(C1)/sqrt(length(C1)) [1] 56.18807 Ex. Toss n = 100 biased coins with P (H) = p. Sup- pose you observe 38 heads. Construct 95% CI of p. > X<-rbinom(100,1,0.4) > X [1] 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 [17] 0 0 1 1 0 1 0 1 1 0 1 0 1 0 1 0 [33] 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 [49] 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 [65] 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 [81] 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 1 [97] 0 1 1 0 > sd(X) [1] 0.4878317 > sqrt(0.38*(1-0.38)/(100-1)) [1] 0.04878317 > 0.38+1.96*0.049/sqrt(100) [1] 0.389604 > 0.38-1.96*0.049/sqrt(100) [1] 0.370396 Self-study problems Example 7.8., Exercise 7.13., 7.19., 7.25. In the above coin tossing example, check if X̄ is MLE.