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Unformatted text preview: STAT511 — Summer 2009 Lecture Notes 1 Chapter 8 July 26, 2009 Statistical Inference 8.1 Hypothesis Testing: The Basics Chapter Overview • Introduction to hypothesis testing – Null hypothesis and alternative hypothesis – Decision procedures – Type I and type II errors • Tests about population mean – Normal with known σ – Normal with unknown σ – Large sample from unknown distribution • Test about population proportion – Sample size n is small – Sample size n is large Statistical Hypothesis • Statistical Hypothesis is a claim about population characteristics, including parameters. In testing a hypothesis, there are two contradictory hypotheses: – Null Hypothesis H : a claim initially assumed to be true, a prior belief, ”old theory”. – Alternative Hypothesis H a : the contradictory claim, the competing claim, ”new theory”. • Decision Principle: reject H in favor of H a if sample data show strong evidence that H is false. Otherwise fail to reject H . • Test of hypothesis is a method for using sample data to decide whether H is to be rejected. Test Procedure A test procedure is specified by the following: • Test statistic ,a function of sample data on which the decision (reject H or do not reject H ) is to be based. • A rejection region , the set of all test statistic values for which H will be rejected. Decision rule: The null hypothesis will be rejected if and only if the observed computed test statistic value falls in the reject region. Question: will we make a wrong decision? Purdue University Chapter8˙print.tex; Last Modified: July 26, 2009 (W. Sharabati) STAT511 — Summer 2009 Lecture Notes 2 Errors in Hypothesis Testing There are two types of errors in hypothesis testing: Definition 1. A type I error consists of rejecting H when H is true. A type II error involves not rejecting H when H is false. Usually, we denote: α = P (type I error) β = P (type II error) Significance Level We want α and β to be both small but this is a contradiction. • We can decrease reject region to get smaller α . • A small reject region result in a larger β . Proposition. When experiment and sample size are fixed and a test statistic is chosen. Decreasing the size of the rejection region to obtain smaller α results in a larger value of β . Type I error is usually more serious than type II error. So we usually select the largest α that can be tolerated and find the appropriate rejection region, α is then called significance level . 8.2 Inferences About the Population Mean Tests about population mean: Normal with Known σ Look at the following hypotheses: • Upper-tailed test: H : μ = μ vs. H a : μ > μ • Lower-tailed test: H : μ = μ vs. H a : μ < μ • Two-tailed test: H : μ = μ vs. H a : μ 6 = μ Upper-tailed Test A random sample X 1 ,X 2 ,...,X n from N ( μ,σ 2 ) with σ 2 known....
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This note was uploaded on 03/30/2011 for the course STAT 511 taught by Professor Bud during the Spring '08 term at Purdue.
- Spring '08