Large Sample Confidence Intervals - Lecture Notes | STAT 312, Study notes of Mathematical Statistics

Download Large Sample Confidence Intervals - Lecture Notes | STAT 312 and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 312: Lecture 08 Large sample confidence intervals Moo K. Chung mchung@stat.wisc.edu September 27, 2004 1. The sample size is inversely related to the width of confidence interval. Example 7.4. 2. Central Limit Theorem. Let X1, · · · , Xn be a random sample with mean µ and variance σ2. For sufficiently large n, Z = X̄ − µ σ/ √ n ∼ N(0, 1). 3. Let X1, · · · , Xn be a random sample with mean µ. For sufficiently large n, Z = X̄ − µ S/ √ n ∼ N(0, 1) where S is the sample standard deviation. If n is sufficiently large, approximate 100(1 − α)% confidence interval for µ is x̄± zα/2 s√ n , where s is the sample standard deviation. 4. General large sample confidence interval. Sup- pose θ̂ is an unbiased estimator of some parame- ter θ, Then 100(1− α)% confidence interval is θ̂ + zα/2 √ Vθ̂. In many applications, Vθ̂ is a function of θ which makes computation of CI complicated. In this sit- uation, we need to estimate Vθ̂ further. Example. Toss n = 100 biased coins with P (H) = p. Suppose you observe 38 heads. Con- struct 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 > sqrt(0.38*(1-0.38)/100)*1.96 [1] 0.09513574 > 0.38+0.095 [1] 0.475 > 0.38-0.095 [1] 0.285 5. One-sided confidence interval: An 100(1 − α)% upper confidence bound for θ is θ < x̄ + zα √ Vθ̂ and a lower confidence bound for µ is θ > x̄− zα √ Vθ̂. Review Problems. Example 7.8, 7.10.