Mathematical Statistics

Mathematics

math448

New York University

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Math 448 Final May 7 Name: ID number: Instructions: • Read problems very carefully. • The correct final answer alone is not sufficient for full credit. Because this is an at home exam, you have more time to explain your answers clearly. • If your answers are not fully explained, they will not receive full credit. • Answer at most one question per piece and scan or take a picture to upload to gradescope. Label each problem on gradescope. PROBLEMS THAT ARE NOT CORRECTLY LABELED AND ORIENTED MAY NOT BE GRADED! • Your final answers need to be simplified only if this is required in the statement of the problem. Otherwise, you there is no need to simplify numerical quantities like 0.3+ 3 4 · 0.3 · 7 + 234 or to reduce fractions like 36 to lowest terms. There is also no need to 5 547  simplify factorials such as 5! or binomial coefficients such as 73 . • You can use any relevant hypothesis that we discussed in class, unless the problem explicitly asks to derive a test. • When computing p-values, state what you use to compute them, you can also just give bounds on the p-value by using the tables in the book. Resources used: I have read the above instructions and the work in this exam is solely my own: Signature: Question: 1 2 3 4 5 Total Points: 10 10 10 10 10 50 Score: 1. (10 points) An infectious disease has 2 mutations, and researchers are interested in what proportion of the population has each of the mutations. They took a random sample of n people and checked if they had mutation 1, mutation 2, or no disease. Let pi be the proportion of the population with mutation i, for i = 1, 2. Recall: If X1 is the number of people with mutation 1 and X2 is the number of people with mutation 2, then the joint distribution of X1 , X2 will be multinomial, that is:   n pX1 ,X2 (x1 , x2 ) = px1 px2 (1 − p1 − p2 )n−x1 −x2 x1 , x2 , n − x1 − x2 1 2 where  n x1 , x2 , n − x1 − x2  = n! x1 !x2 !(n − x1 − x2 )! (a) Derive the Large Sample Size Likelihood Ratio test for H0 : p1 = p2 against Ha : p1 6= p2 , in the large sample regime. Be sure to indicate what Ω0 and Ω are, as well what is the maximum value of the Likelihood Function on these sets. Finding the values where the maximum occurs might look complicated, but this location is somewhat intuitive. (b) If they test 100 people and find 18 have mutation 1, 8 have mutation 2, and the remaining people don’t have the disease. Is there sufficient evidence to claim the proportions are different at α = .05? What is the p-value? 2. (10 points) You have 2 coins, Coin 1 and Coin 2, that each have unknown probability of showing heads after being flipped. Out of the 200 tosses, the Coin 1 shows heads 37 times, and the Coin 2 shows heads 44 times. Let p1 be the probability that the Coin 1 shows heads, and p2 be the probability that Coin 2 shows heads. (a) Construct an approximate 99% confidence interval for p1 − p2 . (b) Does the experiment provide sufficient evidence to indicate that the p1 and p2 are different? Use an α = 0.01 level. (c) Use the test statistic given by the difference in the number of heads in each trial to hypothesis test Ho : p1 = p2 versus Ha : p1 6= p2 with α = 0.01. (d) What is the p-value associated with this test? Page 2 3. (10 points) Suppose Y is a random sample of size 1 from a population with density function ( θ−1 x e−x , for x ≥ 0 Γ(θ) f (y|θ) = 0, otherwise where θ > 0 is an unknown parameter. Based on the single observation of Y , find the uniformly most powerful test at level α for testing H0 : θ = 1 versus Ha : θ < 1. Explain where you use that Ha is a one-sided test. Page 3 4. (10 points) A professor wants to make an exam that is a good assessment of the students ranking, so they want the distribution for all students to have a mean of 75 and standard deviation a of 9. The test is given to 5 randomly selected students, and their scores are: 71, 67, 59, 89, and 93 out of 100. Assume the distribution of students scores is normal. (a) What is the p-value for the test of the null hypothesis H0 : µ = 75 against the alternative hypothesis Ha : µ 6= 75. (b) What is the p-value for the test of the null hypothesis H0 : σ 2 = 92 against the alternative hypothesis Ha : σ 2 > 92 . (c) Give a 2-sided .95-confidence interval for µ. (d) Give am upper tail .95-confidence interval for σ 2 . What should the professor conclude? Page 4 5. (10 points) You want to determine if the percentage of a population that has a disease is greater than 30%, but have very limited resources. So you propose the following test: people are randomly selected from the population and tested until someone tests positive. Once someone tests positive the experiment is stopped and the total number of people tested is recorded. Let p be the total proportion of the population that has has the disease. We test the null hypothesis that H0 : p = .3 against the alternative hypothesis Ha : p > .3. If 3 or less people were tested you reject H0 in favor of Ha . (a) In words, what is a type I error for this test? (b) In words, what is a type II error for this test? (c) What is α, the probability of a type I error? (d) What is β, the probability of a type II error, if p = .4? (e) What is the power of this test as a function of p? Page 5 Continuous Distributions Mean Variance MomentGenerating Function 1 ; θ ≤ y ≤ θ2 θ2 − θ1 1 θ1 + θ2 2 (θ2 − θ1 )2 12 etθ2 − etθ1 t (θ2 − θ1 )     1 1 2 (y − µ) √ exp − 2σ 2 σ 2π −∞ < y < +∞ µ σ2 β β2 (1 − βt)−1 αβ αβ 2 (1 − βt)−α v 2v (1−2t)−v/2 α α+β (α + β) (α + β + 1) Distribution Uniform Normal Exponential Probability Function f (y) = f (y) = f (y) =  Gamma Chi-square f (y) =  1 α−1 −y/β e ; α y (α)β 00 β 0 0 ;  (α + β) y α−1 (1 − y)β−1 ; (α)(β) 0 r Poisson Negative binomial λ y e−λ ; y! y = 0, 1, 2, . . . p(y) = p(y) =  y−1 r−1  p r (1 − p) y−r ; y = r, r + 1, . . . λ λ r p r(1 − p) p 2 exp[λ(et − 1)]  pet 1 − (1 − p)et r MATHEMATICAL STATISTICS WITH APPLICATIONS This page intentionally left blank SEVENTH EDITION Mathematical Statistics with Applications Dennis D. Wackerly University of Florida William Mendenhall III University of Florida, Emeritus Richard L. Scheaffer University of Florida, Emeritus Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States Mathematical Statistics with Applications, Seventh Edition Dennis D. Wackerly, William Mendenhall III, Richard L. Scheaffer Statistics Editor: Carolyn Crockett Assistant Editors: Beth Gershman, Catie Ronquillo Editorial Assistant: Ashley Summers Technology Project Manager: Jennifer Liang Marketing Manager: Mandy Jellerichs Marketing Assistant: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent Project Manager, Editorial Production: Hal Humphrey Art Director: Vernon Boes Print Buyer: Karen Hunt Production Service: Matrix Productions Inc. Copy Editor: Betty Duncan Cover Designer: Erik Adigard, Patricia McShane Cover Image: Erik Adigard Cover Printer: TK Compositor: International Typesetting and Composition Printer: TK © 2008, 2002 Duxbury, an imprint of Thomson Thomson Higher Education 10 Davis Drive Belmont, CA 94002-3098 USA Brooks/Cole, a part of The Thomson Corporation. Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license. ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher. Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 09 08 07 ExamView® and ExamView Pro® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. © 2008 Thomson Learning, Inc. All Rights Reserved. Thomson Learning WebTutorTM is a trademark of Thomson Learning, Inc. International Student Edition ISBN-13: 978-0-495-38508-0 ISBN-10: 0-495-38508-5 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com. Any additional questions about permissions can be submitted by e-mail to thomsonrights@thomson.com. CONTENTS Preface xiii Note to the Student xxi 1 What Is Statistics? 1 1.1 Introduction 1.2 Characterizing a Set of Measurements: Graphical Methods 3 1.3 Characterizing a Set of Measurements: Numerical Methods 8 1.4 How Inferences Are Made 1.5 Theory and Reality 1.6 Summary 1 13 14 15 2 Probability 20 2.1 Introduction 2.2 Probability and Inference 21 2.3 A Review of Set Notation 23 2.4 A Probabilistic Model for an Experiment: The Discrete Case 2.5 Calculating the Probability of an Event: The Sample-Point Method 2.6 Tools for Counting Sample Points 2.7 Conditional Probability and the Independence of Events 2.8 Two Laws of Probability 20 26 35 40 51 57 v vi Contents 2.9 Calculating the Probability of an Event: The Event-Composition Method 62 2.10 The Law of Total Probability and Bayes’ Rule 2.11 Numerical Events and Random Variables 2.12 Random Sampling 2.13 Summary 70 75 77 79 3 Discrete Random Variables and Their Probability Distributions 86 3.1 Basic Definition 3.2 The Probability Distribution for a Discrete Random Variable 3.3 The Expected Value of a Random Variable or a Function of a Random Variable 91 3.4 The Binomial Probability Distribution 3.5 The Geometric Probability Distribution 3.6 The Negative Binomial Probability Distribution (Optional) 121 3.7 The Hypergeometric Probability Distribution 3.8 The Poisson Probability Distribution 3.9 Moments and Moment-Generating Functions 138 3.10 Probability-Generating Functions (Optional) 143 3.11 Tchebysheff’s Theorem 3.12 Summary 86 87 100 114 125 131 146 149 4 Continuous Variables and Their Probability Distributions 157 4.1 Introduction 4.2 The Probability Distribution for a Continuous Random Variable 4.3 Expected Values for Continuous Random Variables 4.4 The Uniform Probability Distribution 4.5 The Normal Probability Distribution 178 4.6 The Gamma Probability Distribution 185 4.7 The Beta Probability Distribution 157 174 194 170 158 Contents vii 4.8 Some General Comments 4.9 Other Expected Values 4.10 Tchebysheff’s Theorem 4.11 Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional) 210 4.12 Summary 201 202 207 214 5 Multivariate Probability Distributions 223 5.1 Introduction 5.2 Bivariate and Multivariate Probability Distributions 224 5.3 Marginal and Conditional Probability Distributions 235 5.4 Independent Random Variables 5.5 The Expected Value of a Function of Random Variables 5.6 Special Theorems 5.7 The Covariance of Two Random Variables 5.8 The Expected Value and Variance of Linear Functions of Random Variables 270 5.9 The Multinomial Probability Distribution 5.10 The Bivariate Normal Distribution (Optional) 5.11 Conditional Expectations 5.12 Summary 223 247 255 258 264 279 283 285 290 6 Functions of Random Variables 296 6.1 Introduction 6.2 Finding the Probability Distribution of a Function of Random Variables 297 6.3 The Method of Distribution Functions 6.4 The Method of Transformations 6.5 The Method of Moment-Generating Functions 6.6 Multivariable Transformations Using Jacobians (Optional) 6.7 Order Statistics 6.8 Summary 296 341 333 298 310 318 325 viii Contents 7 Sampling Distributions and the Central Limit Theorem 346 7.1 Introduction 7.2 Sampling Distributions Related to the Normal Distribution 7.3 The Central Limit Theorem 7.4 A Proof of the Central Limit Theorem (Optional) 7.5 The Normal Approximation to the Binomial Distribution 7.6 Summary 346 353 370 377 378 385 8 Estimation 390 8.1 Introduction 8.2 The Bias and Mean Square Error of Point Estimators 8.3 Some Common Unbiased Point Estimators 8.4 Evaluating the Goodness of a Point Estimator 8.5 Confidence Intervals 8.6 Large-Sample Confidence Intervals 8.7 Selecting the Sample Size 8.8 Small-Sample Confidence Intervals for µ and µ1 − µ2 8.9 Confidence Intervals for σ 8.10 Summary 390 392 396 399 406 411 421 2 425 434 437 9 Properties of Point Estimators and Methods of Estimation 444 9.1 Introduction 9.2 Relative Efficiency 9.3 Consistency 9.4 Sufficiency 9.5 The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation 464 9.6 The Method of Moments 9.7 The Method of Maximum Likelihood 9.8 Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional) 483 9.9 Summary 444 445 448 459 485 472 476 Contents ix 10 Hypothesis Testing 488 10.1 Introduction 10.2 Elements of a Statistical Test 10.3 Common Large-Sample Tests 10.4 Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests 507 10.5 Relationships Between Hypothesis-Testing Procedures and Confidence Intervals 511 10.6 Another Way to Report the Results of a Statistical Test: Attained Significance Levels, or p-Values 513 10.7 Some Comments on the Theory of Hypothesis Testing 10.8 Small-Sample Hypothesis Testing for µ and µ1 − µ2 10.9 Testing Hypotheses Concerning Variances 10.10 Power of Tests and the Neyman–Pearson Lemma 10.11 Likelihood Ratio Tests 10.12 Summary 488 489 496 518 520 530 540 549 556 11 Linear Models and Estimation by Least Squares 563 11.1 Introduction 11.2 Linear Statistical Models 11.3 The Method of Least Squares 11.4 Properties of the Least-Squares Estimators: Simple Linear Regression 577 11.5 Inferences Concerning the Parameters βi 11.6 Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression 589 11.7 Predicting a Particular Value of Y by Using Simple Linear Regression 593 11.8 Correlation 11.9 Some Practical Examples 11.10 Fitting the Linear Model by Using Matrices 11.11 Linear Functions of the Model Parameters: Multiple Linear Regression 615 11.12 Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression 616 564 566 569 584 598 604 609 x Contents 11.13 Predicting a Particular Value of Y by Using Multiple Regression 11.14 A Test for H0 : βg+1 = βg+2 = · · · = βk = 0 11.15 Summary and Concluding Remarks 622 624 633 12 Considerations in Designing Experiments 640 12.1 The Elements Affecting the Information in a Sample 12.2 Designing Experiments to Increase Accuracy 12.3 The Matched-Pairs Experiment 12.4 Some Elementary Experimental Designs 12.5 Summary 640 641 644 651 657 13 The Analysis of Variance 661 13.1 Introduction 13.2 The Analysis of Variance Procedure 13.3 Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout 667 13.4 An Analysis of Variance Table for a One-Way Layout 13.5 A Statistical Model for the One-Way Layout 13.6 Proof of Additivity of the Sums of Squares and E(MST) for a One-Way Layout (Optional) 679 13.7 Estimation in the One-Way Layout 13.8 A Statistical Model for the Randomized Block Design 13.9 The Analysis of Variance for a Randomized Block Design 13.10 Estimation in the Randomized Block Design 13.11 Selecting the Sample Size 13.12 Simultaneous Confidence Intervals for More Than One Parameter 13.13 Analysis of Variance Using Linear Models 13.14 Summary 661 662 671 677 681 686 688 695 696 701 705 14 Analysis of Categorical Data 713 14.1 A Description of the Experiment 14.2 The Chi-Square Test 14.3 A Test of a Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test 716 713 714 698 Contents xi 14.4 Contingency Tables 14.5 r × c Tables with Fixed Row or Column Totals 14.6 Other Applications 14.7 Summary and Concluding Remarks 721 729 734 736 15 Nonparametric Statistics 741 15.1 Introduction 15.2 A General Two-Sample Shift Model 15.3 The Sign Test for a Matched-Pairs Experiment 15.4 The Wilcoxon Signed-Rank Test for a Matched-Pairs Experiment 15.5 Using Ranks for Comparing Two Population Distributions: Independent Random Samples 755 15.6 The Mann–Whitney U Test: Independent Random Samples 15.7 The Kruskal–Wallis Test for the One-Way Layout 15.8 The Friedman Test for Randomized Block Designs 15.9 The Runs Test: A Test for Randomness 15.10 Rank Correlation Coefficient 15.11 Some General Comments on Nonparametric Statistical Tests 741 742 744 765 771 777 783 16 Introduction to Bayesian Methods for Inference 796 16.1 Introduction 16.2 Bayesian Priors, Posteriors, and Estimators 16.3 Bayesian Credible Intervals 16.4 Bayesian Tests of Hypotheses 16.5 Summary and Additional Comments 796 797 808 813 816 Appendix 1 Matrices and Other Useful Mathematical Results 821 A1.1 Matrices and Matrix Algebra A1.2 Addition of Matrices A1.3 Multiplication of a Matrix by a Real Number A1.4 Matrix Multiplication 821 822 823 758 823 789 750 xii Contents A1.5 Identity Elements A1.6 The Inverse of a Matrix A1.7 The Transpose of a Matrix A1.8 A Matrix Expression for a System of Simultaneous Linear Equations 828 A1.9 Inverting a Matrix A1.10 Solving a System of Simultaneous Linear Equations A1.11 Other Useful Mathematical Results 825 827 828 830 834 835 Appendix 2 Common Probability Distributions, Means, Variances, and Moment-Generating Functions 837 Table 1 Discrete Distributions 837 Table 2 Continuous Distributions 838 Appendix 3 Tables 839 Table 1 Binomial Probabilities Table 2 Table of e−x Table 3 Poisson Probabilities 843 Table 4 Normal Curve Areas 848 Table 5 Percentage Points of the t Distributions Table 6 Percentage Points of the χ 2 Distributions Table 7 Percentage Points of the F Distributions Table 8 Distribution Function of U Table 9 Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50 868 839 842 849 850 852 862 Table 10 Distribution of the Total Number of Runs R in Samples of Size (n 1 , n 2 ); P(R ≤ a) 870 Table 11 Critical Values of Spearman’s Rank Correlation Coefficient Table 12 Random Numbers Answers to Exercises Index 896 877 873 872 PREFACE The Purpose and Prerequisites of this Book Mathematical Statistics with Applications was written for use with an undergraduate 1-year sequence of courses (9 quarter- or 6 semester-hours) on mathematical statistics. The intent of the text is to present a solid undergraduate foundation in statistical theory while providing an indication of the relevance and importance of the theory in solving practical problems in the real world. We think a course of this type is suitable for most undergraduate disciplines, ...
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