Beginning Statistics - 2012 Book Archive

716 Pages·2012·33.31 MB·English

Beginning Statistics
v. 1.0
This is the bookBeginning Statistics(v. 1.0).
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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
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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
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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.
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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
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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.
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Dedication
To our families and teachers.
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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,
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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.
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