Duality Between Tests and Confidence Intervals - Lecture Slides | STAT 703, Study notes of Statistics

Download Duality Between Tests and Confidence Intervals - Lecture Slides | STAT 703 and more Study notes Statistics in PDF only on Docsity! 1 STAT 703/J703 B.Habing Univ. of SC 1 STAT 703/J703 March 6th, 2007 -Lecture 15- Instructor: Brian Habing Department of Statistics LeConte 203 Telephone: 803-777-3578 E-mail: habing@stat.sc.edu STAT 703/J703 B.Habing Univ. of SC 2 Today • Homework 4 • 9.3: Duality Between Tests and Confidence Intervals • 9.2.1: The p-value • Power of a Test STAT 703/J703 B.Habing Univ. of SC 3 Chapter 9: #7) Let X1, … Xn be a sample from a Poisson distribution. Find the likelihood ratio for testing H0: λ=λ0 versus HA: λ=λ1 where λ1>λ0. Explain how to determine a rejection region for a test at level α. 2 STAT 703/J703 B.Habing Univ. of SC 4 Chapter 9: #8) Show that the test of problem 7 is UMP for H0: λ=λ0 versus HA: λ> λ0. Consider a sample of size 20 and the null hypothesis λ =1. Find the possible alpha level closest to 0.05 without going exceeding it, and the corresponding rejection region. STAT 703/J703 B.Habing Univ. of SC 5 9.3 The Duality of Confidence Intervals and Hypothesis Tests There is a duality between confidence intervals and hypothesis tests. A confidence interval is found by “inverting” a two-sided test (and vice-versa). STAT 703/J703 B.Habing Univ. of SC 6 Theorem A, pg. 338: Suppose there is a test of level α for H0: θ= θ0, and let A(θ0)=acceptance region Then the set C={θ: X∈A(θ)} is a 100(1- α)% confidence region for θ.