INTRO STAT THEORY II
INTRO STAT THEORY II STAT 703
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This 6 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 703 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/229670/stat-703-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.
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Date Created: 10/26/15
STAT 703J703 February 8th 2005 Lecture 9 lnstructor Brian Habing Department of Statistics LeConte 203 Telephone 8037773578 Email habingstatscedu STAT 7amp31703 BHabing Univ of SC Today Homework 3 Solutions Basics of Hypothesis Testing Chapter 9 STAT 7amp31703 BHabing Univ of SC w Ch85 Suppose that X follows a geometric distribution PX k Iva p And assume an iid sample of size n b Find the mle of 0 STAT 7amp31703 BHabing Univ of SC w c Find the asymptotic variance of the MLE v STAT 7amp31703 BHabing Univ of SC M Ch86 Consider the data Hops Freg Hops Freg 1 48 7 4 2 31 8 2 3 20 9 1 4 9 10 1 5 6 11 2 6 5 12 1 a Fit a geometric distribution b Find an approximate 95 confidence interval for p v STAT 7amp31703 BHabing Univ of SC M Ch844c Let X1Xn be iid random variables with density function fx661x 90 x 1 Find the asymptotic variance of the MLE v STAT 7amp31703 BHabing Univ of SC M Statistical Inference Confidence Intervals and Tests of Hypotheses Test of hypothesis general method to distinguish between 2 or more probability distributions or models based on a sample X1 Xn assumed to come from one of them v STAT 7amp31703 BHabing Univ of SC 4 M In particular based on X1 Xn decide whether f1x or f2x is the pdf or population from which the sample came More specifically suppose we think the sample is from a normal population with mean either u5 or u10 with variance 4 Or more generally uu1 vs uu2 STAT 7amp31703 BHabing Univ of SC M To do this we follow the Neyman Pearson Paradigm and discuss significance tests NeymanPearson Approach Group the possible hypothesized distributions into two categories the null hypothesis HO and the alternative h othesis HA research hypothesis V STAT703J703 BHabing Univ ofSC 10 Lg Observe X1 Xn Either H0 Np1 62 or HA Np2 62 or H0 pp1 vs HA pp2 p is the mean of Np 62 Here if 02 is known each of these hypotheses completer specifies the distribution or population 80 HO and HA are called simple hypotheses v STAT 7amp31703 BHabing Univ of SC 4 w n If H0 u0 VS HA u gt 0 u in Nu 62 with 02 known Then H0 is simple and HA is a composite hypothesis ie several normal distributions would satisfy it HA is also referred to as a onesided hypothesis v STAT 7amp31703 BHabing Univ of SC 12 M If H0 u0 VS HA who u in Nu 62 02 known then HA is a twosided composite hypothesis Next we set up the framework for testing HO and HA based on the sample v STAT 7amp31703 BHabing Univ of SC 13 M 92 NevmanPearson Paradigm Let x X1 Xn denote a sample from population fxe Decide on HO 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 TTXTX1 Xn w STAT 7amp31703 BHabing Univ of SC 14 M 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 reiection region of the test v STAT 7amp31703 BHabing Univ of SC 15 M Two kinds of error may occur 1 H0 is rejected when it is true Type I error Ptype I error or PT e rejection region HO true If H0 is simple or is called the significance level of the test v STAT 7amp31703 BHabing Univ of SC 16 M 2 H0 is accepted when it is false Type II error Ptype ll error 3 PT in acceptance region HO false If HA is composite 3 depends on which member of HA is the true pdf v STAT 703J703 BHabing Univ of SC 4 w 17 Power of the test PHO is rejected when false 1 PHO is accepted HO false 1 3 Ideally we would want oc30 but this not possible since the decision is based on data v STAT 703J703 BHabing Univ of SC 18 M
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