Binomial Distributions: Point and Confidence Interval Estimation and Prediction, Study notes of Statistics

Download Binomial Distributions: Point and Confidence Interval Estimation and Prediction and more Study notes Statistics in PDF only on Docsity! STATISTICS 301 TA: Perla E. Reyes DISCUSSION 7 Pag. 1 Review Estimation: Suppose X has a binomial distribution, that is X ∼ Bin(n, p): 1. Point Estimation We collectED n observations and obtainED x successes, then p̂ = x n q̂ = 1− p̂ = 1− x n 2. Confidence Interval Estimation We collectED n observations and obtainED x successes, then p̂± z √ p̂q̂ n where h = z √ p̂q̂ n is called the half −width The most popular choice for confidence level is 95%, the value of z for this and other choices of confidence levels are: Confidence Level z 80% 1.282 90% 1.645 95% 1.960 98% 2.326 99% 2.576 3. Confidence Interval Interpretation • A 95% confidence interval for p means that if we conduct the same experiments again and again and again, and get a bunch of C.I.s ( NOTE : these C.I.s may be different), for 100 times, about 95 times the confidence intervals will cover the true value of p. • A confidence interval for p is called correct if it contains the true value of p. • A higher confidence level achieves its higher chance of being correct at the expense of producing a wider interval . A wider confidence interval is less useful. Suppose that we don’t want to make any mistake, we can use [0,1] to be our C.I. since it’s confidence level is 100%. However, we know p is always between 0 and 1. What advantage can we get from this kind of C.I.? • For fixed p̂, the larger the value of n , the smaller the value of h. • For fixed p̂, the larger the value of the confidence level, the larger the value of h. • For fixed n, as the value of p̂ moves away from 0.5, in either direction, h decreases. reyes@stat.wisc.edu. www.stat.wisc.edu/∼reyes/ B248MSC, MW 11:00-12:00