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You can browse or download additional books there. ii Table of Contents About the Authors................................................................................................................. 1 Acknowledgements............................................................................................................... 2 Dedication............................................................................................................................... 4 Preface..................................................................................................................................... 5 Chapter 1: Introduction....................................................................................................... 7 Basic Definitions and Concepts.....................................................................................................................8 Overview.......................................................................................................................................................16 Presentation of Data....................................................................................................................................18 Chapter 2: Descriptive Statistics ..................................................................................... 21 Three Popular Data Displays.......................................................................................................................22 Measures of Central Location.....................................................................................................................38 Measures of Variability...............................................................................................................................57 Relative Position of Data.............................................................................................................................69 The Empirical Rule and Chebyshev’s Theorem.........................................................................................90 Chapter 3: Basic Concepts of Probability..................................................................... 110 Sample Spaces, Events, and Their Probabilities.....................................................................................111 Complements, Intersections, and Unions................................................................................................130 Conditional Probability and Independent Events..................................................................................153 Chapter 4: Discrete Random Variables ........................................................................ 180 Random Variables......................................................................................................................................181 Probability Distributions for Discrete Random Variables.....................................................................185 The Binomial Distribution.........................................................................................................................206 Chapter 5: Continuous Random Variables................................................................... 227 Continuous Random Variables.................................................................................................................228 The Standard Normal Distribution..........................................................................................................242 Probability Computations for General Normal Random Variables......................................................254 Areas of Tails of Distributions..................................................................................................................269 Chapter 6: Sampling Distributions................................................................................ 287 The Mean and Standard Deviation of the Sample Mean........................................................................288 The Sampling Distribution of the Sample Mean.....................................................................................293 The Sample Proportion.............................................................................................................................311 iii Chapter 7: Estimation ...................................................................................................... 325 Large Sample Estimation of a Population Mean.....................................................................................326 Small Sample Estimation of a Population Mean.....................................................................................340 Large Sample Estimation of a Population Proportion............................................................................351 Sample Size Considerations......................................................................................................................362 Chapter 8: Testing Hypotheses....................................................................................... 372 The Elements of Hypothesis Testing........................................................................................................373 Large Sample Tests for a Population Mean.............................................................................................388 The Observed Significance of a Test........................................................................................................400 Small Sample Tests for a Population Mean.............................................................................................413 Large Sample Tests for a Population Proportion....................................................................................427 Chapter 9: Two-Sample Problems ................................................................................. 443 Comparison of Two Population Means: Large, Independent Samples.................................................444 Comparison of Two Population Means: Small, Independent Samples.................................................466 Comparison of Two Population Means: Paired Samples........................................................................481 Comparison of Two Population Proportions...........................................................................................500 Sample Size Considerations......................................................................................................................519 Chapter 10: Correlation and Regression...................................................................... 531 Linear Relationships Between Variables.................................................................................................532 The Linear Correlation Coefficient...........................................................................................................541 Modelling Linear Relationships with Randomness Present..................................................................555 The Least Squares Regression Line..........................................................................................................560 Statistical Inferences About ?1.................................................................................................................581 The Coefficient of Determination.............................................................................................................596 Estimation and Prediction........................................................................................................................606 A Complete Example..................................................................................................................................618 Formula List................................................................................................................................................629 Chapter 11: Chi-Square Tests and F-Tests................................................................... 631 Chi-Square Tests for Independence.........................................................................................................632 Chi-Square One-Sample Goodness-of-Fit Tests.......................................................................................650 F-tests for Equality of Two Variances......................................................................................................664 F-Tests in One-Way ANOVA......................................................................................................................687 Appendix............................................................................................................................. 702 iv About the Authors Douglas S. Shafer Douglas Shafer is Professor of Mathematics at the University of North Carolina at Charlotte. In addition to his position in Charlotte he has held visiting positions at the University of Missouri at Columbia and Montana State University and a Senior Fulbright Fellowship in Belgium. He teaches a range of mathematics courses as well as introductory statistics. In addition to journal articles and this statistics textbook, he has co-authored with V. G. Romanovski (Maribor, Slovenia) a graduate textbook in his research specialty. He earned a PhD in mathematics at the University of North Carolina at Chapel Hill. Zhiyi Zhang Zhiyi Zhang is Professor of Mathematics at the University of North Carolina at Charlotte. In addition to his teaching and research duties at the university, he consults actively to industries and governments on a wide range of statistical issues. His research activities in statistics have been supported by National Science Foundation, U.S. Environmental Protection Agency, Office of Naval Research, and National Institute of Health. He earned a PhD in statistics at Rutgers University in New Jersey. 1 Acknowledgements We would like to thank the following colleagues whose comprehensive feedback and suggestions for improving the material helped us make a better text: Kathy Autrey, Northwestern State University Kiran Bhutani, The Catholic University of America Rhonda Buckley, Texas Woman’s University Susan Cashin, University of Wisconsin-Milwaukee Kathryn Cerrone, The University of Akron-Summit College Zhao Chen, Florida Gulf Coast University Ilhan Izmirli, George Mason University, Department of Statistics Denise Johansen, University of Cincinnati Eric Kean, Western Washington University Yolanda Kumar, Univeristy of Missouri-Columbia Eileen Stock, Baylor University Sean Thomas, Emory University Sara Tomek, University of Alabama Mildred Vernia, Indiana University Southeast 2 Acknowledgements Gingia Wen, Texas Woman’s University Jiang Yuan, Baylor University We also acknowledge the valuable contribution of the publisher’s accuracy checker, Phyllis Barnidge. 3 Dedication To our families and teachers. 4 Preface This book is meant to be a textbook for a standard one-semester introductory statistics course for general education students. Our motivation for writing it is twofold: 1.) to provide a low-cost alternative to many existing popular textbooks on the market; and 2.) to provide a quality textbook on the subject with a focus on the core material of the course in a balanced presentation. The high cost of textbooks has spiraled out of control in recent years. The high frequency at which new editions of popular texts appear puts a tremendous burden on students and faculty alike, as well as the natural environment. Against this background we set out to write a quality textbook with materials such as examples and exercises that age well with time and that would therefore not require frequent new editions. Our vision resonates well with the publisher’s business model which includes free digital access, reduced paper prints, and easy customization by instructors if additional material is desired. Over time the core content of this course has developed into a well-defined body of material that is substantial for a one-semester course. The authors believe that the students in this course are best served by a focus on the core material and not by an exposure to a plethora of peripheral topics. Therefore in writing this book we have sought to present material that comprises fully a central body of knowledge that is defined according to convention, realistic expectation with respect to course duration and students’ maturity level, and our professional judgment and experience. We believe that certain topics, among them Poisson and geometric distributions and the normal approximation to the binomial distribution (particularly with a continuity correction) are distracting in nature. Other topics, such as nonparametric methods, while important, do not belong in a first course in statistics. As a result we envision a smaller and less intimidating textbook that trades some extended and unnecessary topics for a better focused presentation of the central material. Textbooks for this course cover a wide range in terms of simplicity and complexity. Some popular textbooks emphasize the simplicity of individual concepts to the point of lacking the coherence of an overall network of concepts. Other textbooks include overly detailed conceptual and computational discussions and as a result repel students from reading them. The authors believe that a successful book must strike a balance between the two extremes, however difficult it may be. As a consequence the overarching guiding principle of our writing is to seek simplicity but to preserve the coherence of the whole body of information communicated, 5 Preface both conceptually and computationally. We seek to remind ourselves (and others) that we teach ideas, not just step-by-step algorithms, but ideas that can be implemented by straightforward algorithms. In our experience most students come to an introductory course in statistics with a calculator that they are familiar with and with which their proficiency is more than adequate for the course material. If the instructor chooses to use technological aids, either calculators or statistical software such as Minitab or SPSS, for more than mere arithmetical computations but as a significant component of the course then effective instruction for their use will require more extensive written instruction than a mere paragraph or two in the text. Given the plethora of such aids available, to discuss a few of them would not provide sufficiently wide or detailed coverage and to discuss many would digress unnecessarily from the conceptual focus of the book. The overarching philosophy of this textbook is to present the core material of an introductory course in statistics for non-majors in a complete yet streamlined way. Much room has been intentionally left for instructors to apply their own instructional styles as they deem appropriate for their classes and educational goals. We believe that the whole matter of what technological aids to use, and to what extent, is precisely the type of material best left to the instructor’s discretion. All figures with the exception ofFigure 1.1 "The Grand Picture of Statistics",Figure 2.1 "Stem and Leaf Diagram",Figure 2.2 "Ordered Stem and Leaf Diagram",Figure 2.13 "The Box Plot",Figure 10.4 "Linear Correlation Coefficient ",Figure 10.5 "The Simple Linear Model Concept", and the unnumbered figure inNote 2.50 "Example 16"ofChapter 2 "Descriptive Statistics"were generated using MATLAB, copyright 2010. 6