Lecture Notes on Confidence Intervals III | STAT 312, Study notes of Mathematical Statistics

Download Lecture Notes on Confidence Intervals III | STAT 312 and more Study notes Mathematical Statistics in PDF only on Docsity! Stat 312: Lecture 7 Confidence Intervals III. Moo K. Chung mchung@stat.wisc.edu February 11, 2003 Concepts 1. If n is sufficiently large, approximate 100(1 − α)% confidence interval for µ is x̄ ± zα/2s/ √ n, where s is the sample stan- dard deviation. 2. Let p denote the proportion of an individual with a specified property. 100(1 − α)% CI for a population proportion p is p̂ ± zα/2 √ p̂q̂/n. 3. One-sided confidence interval: An upper confidence bound for µ is µ < x̄ + zαs/ √ n and a lower confidence bound for µ is µ > x̄ − zαs/ √ n. In-class problems Ex. Toss n = 100 biased coins with P (H) = p. Suppose you observe 38 heads. Construct 95% CI of p. rbinom(n,1,p) will generate a Bernoulli random sample of size n with P (Xi = 1) = 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 > 0.38+1.96*0.49/sqrt(100) [1] 0.47604 > 0.38-1.96*0.49/sqrt(100) [1] 0.28396 Exercise 7.23. Use Concept 2. When 37 helmets were subjected to a certain im- pact test, 24 showed damage. Let p denote the proportion of helmets that would show damage under the test. Find a 99% CI for p. Self-study problems Example 7.8., 7.8. Use Concept 2.