Download Basics of Hypothesis Testing - Homework 3 with Solutions | 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 February 8th, 2005 -Lecture 9- 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 3 Solutions • Basics of Hypothesis Testing (Chapter 9) STAT 703/J703 B.Habing Univ. of SC 3 Ch.8#5 Suppose that X follows a geometric distribution, And assume an i.i.d. sample of size n. b) Find the mle of p. 1)1(][ −−== kppkXP 2 STAT 703/J703 B.Habing Univ. of SC 4 c) Find the asymptotic variance of the MLE. STAT 703/J703 B.Habing Univ. of SC 5 Ch.8#6 Consider the data # Hops Freq # Hops Freq 1 48 7 4 2 31 8 2 3 20 9 1 4 9 10 1 5 6 11 2 6 5 12 1 STAT 703/J703 B.Habing Univ. of SC 6 a) Fit a geometric distribution b) Find an approximate 95% confidence interval for p. 5 STAT 703/J703 B.Habing Univ. of SC 13 If H0: µ=0 vs. HA: µ≠0 (µ in N(µ, σ2), σ2 known), then HA is a two-sided (composite) hypothesis. Next, we set up the framework for “testing” H0 and HA based on the sample. STAT 703/J703 B.Habing Univ. of SC 14 9.2 Neyman-Pearson Paradigm Let X = (X1, …, Xn) denote a sample from population f(x|θ). Decide on H0 vs. HA based on the sample. A decision on whether or not to reject H0 in favor of HA is made on the basis of a statistic T=T(X)=T(X1, …, Xn). STAT 703/J703 B.Habing Univ. of SC 15 The set of values of T for which H0 is accepted is called the acceptance region and the set of values of T for which H0 is rejected is the rejection region of the test. 6 STAT 703/J703 B.Habing Univ. of SC 16 Two kinds of error may occur: 1. H0 is rejected when it is true: Type I error. P(type I error) = α = P(T ∈ rejection region | H0 true). If H0 is simple, α is called the significance level of the test. STAT 703/J703 B.Habing Univ. of SC 17 2. H0 is accepted when it is false: Type II error. P(type II error) = β = P(T in acceptance region | H0 false) If HA is composite, β depends on which member of HA is the true pdf. STAT 703/J703 B.Habing Univ. of SC 18 Power of the test =P(H0 is rejected when false) = 1 - P(H0 is accepted | H0 false) = 1 - β. Ideally, we would want α=β=0, but this not possible since the decision is based on data.
