Understanding Statistics Using R Randall Schumacker Sara Tomek ● Understanding Statistics Using R Randall Schumacker Sara Tomek University of Alabama University of Alabama Tuscaloosa, AL, USA Tuscaloosa, AL, USA ISBN 978-1-4614-6226-2 ISBN 978-1-4614-6227-9 (eBook) DOI 10.1007/978-1-4614-6227-9 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012956055 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci fi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro fi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speci fi cally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci fi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to our children, Rachel and Jamie Daphne Preface This book was written as a supplemental text for use with introductory or intermedi- ate statistics books. The content of each chapter is appropriate for any undergradu- ate or graduate level statistics course. The chapters are ordered along the lines of many popular statistics books so it should be easy to supplement the chapter content and exercises with your statistics book and lecture materials. The content of each chapter was written to enrich a students’ understanding of statistics using R simula- tion programs. The chapter exercises reinforce an understanding of the statistical concepts presented in the chapters. Computational skills are kept to a minimum in the book by including R script pro- grams that can be run for the exercises in the chapters. Students are not required to master the writing of R script programs, but explanations of how the programs work and program output are included in each chapter. R is a statistical package with an extensive library of functions that offers fl exibility in writing customized statistical routines. The R script commands are run in the R Studio software which is a graphical user interface for Windows. The R Studio software makes accessing R programs, viewing output from the exercises, and graph displays easier for the student. Organization of the Text The fi rst chapter of the book covers fundamentals of R. This includes installation of R and R Studio, accessing R packages and libraries of functions. The chapter also covers how to access manuals and technical documentation, as well as, basic R commands used in the R script programs in the chapters. This chapter is important for the instructor to master so that the software can be installed and the R script programs run. The R software is free permitting students to install the software and run the R script programs for the chapter exercises. The second chapter offers a rich insight into how probability has shaped statistics in the behavioral sciences. This chapter begins with an understanding of fi nite and vii viii Preface in fi nite probability. Key probability concepts related to joint, addition, multiplica- tion, and conditional probability are covered with associated exercises. Finally, the all important combination and permutation concepts help to understand the seven fundamental rules of probability theory which impact statistics. Chapter 3 covers statistical theory as it relates to taking random samples from a population. The R script program is run to demonstrate sampling error. Basically, sampling error is expected to be reduced as size of the random sample increases. Another important concept is the generation of random numbers. Random numbers should not repeat or be correlated when sampling without replacement. Chapter 4 covers histograms and ogives, population distributions, and stem and leaf graphs. The frequency distribution of cumulative percents is an ogive, repre- sented by a characteristic S-shaped curve. In contrast, a data distribution can be unimodal or bimodal, increasing or decreasing in value. A stem and leaf graph fur- ther helps to visualize the data distribution, middle value and range or spread of the data. Graphical display of data is reinforced by the chapter exercises. Chapter 5 covers measures of central tendency and dispersion. The concept of mean and median are presented in the chapter exercises, as well as the concept of dispersion or variance. Sample size effects are then presented to better understand how small versus large samples impact central tendency and dispersion. The Tchebysheff Inequality Theorem is presented to introduce the idea of capturing scores within certain standard deviations of the frequency distribution of data, espe- cially when it is not normally distributed. The normal distribution is presented next followed by the Central Limit Theorem, which provides an understanding that sam- pling distributions will be normally distributed regardless of the shape of the popu- lation from which the random sample was drawn. Chapter 6 covers an understanding of statistical distributions. Binomial distribu- tions formed from the probability or frequency of dichotomous data are covered. The normal distribution is discussed both as a mathematical formula and as proba- bility under the normal distribution. The shape and properties of the chi-square distribution, t-distribution, and F-distribution are also presented. Some basic tests of variance are introduced in the chapter exercises. Chapter 7 discusses hypothesis testing by expressing the notion that “ A statistic is to a sample as a parameter is to a population ”. The concept of a sampling distri- bution is explained as a function of sample size. Con fi dence intervals are introduced for different probability areas of the sampling distribution that capture the popula- tion parameter. The R program demonstrates the con fi dence interval around the sample statistic is computed by using the standard error of the statistic. The statisti- cal hypothesis with null and alternative expressions for percents, ranks, means, and correlation are introduced. The basic idea of testing whether a sample statistic falls outside the null area of probability is demonstrated in the R program. Finally TYPE I and TYPE II error are discussed and illustrated in the chapter exercises using R programs. Chapters 8 – 13 cover the statistics taught in an elementary to intermediate statis- tics course. The statistics covered are chi-square, z, t, F, correlation, and regression. The respective chapters discuss hypothesis testing steps using these statistics. The R Preface ix programs further calculate the statistics and related output for interpretation of results. These chapters form the core content of the book whereas the earlier chap- ters lay the foundation and groundwork for understanding the statistics. A real bene fi t of using the R programs for these statistics is that students have free access at home and school. An instructor can also use the included R functions for the statistics in class thereby greatly reducing any programming or computational time by students. Chapter 1 4 is included to present the concept that research should be replicated to validate fi ndings. In the absence of being able to replicate a research study, the idea of cross validation, jackknife, and bootstrap are commonly used methods. These methods are important to understand and use when conducting research. The R programs make these efforts easy to conduct. Students gain further insight in Chap. 1 5 where a synthesis of research fi ndings help to understand overall what research results indicate on a speci fi c topic. It further illustrates how the statistics covered in the book can be converted to a common scale so that effect size measures can be calculated, which permits the quantitative synthesis of statistics reported in research studies. The chapter concludes by pointing out that statistical signi fi cance testing, i.e., p < 0.05, is not necessarily suf fi cient evidence of the practical impor- tance of research results. It highlights the importance of reporting the sample statis- tic, signi fi cance level, con fi dence interval, and effect size. Reporting of these values extends the students’ thinking beyond signi fi cance testing. R Programs The chapters contain one or more R programs that produce computer output for the chapter exercises. The R script programs enhance the basic understanding and con- cepts in the chapters. The R programs in each chapter are labeled for easy identifi cation. A bene fi t of using the R programs is that the R software is free for home or school use. After mastering the concepts in the book, the R software can be used for data analysis and graphics using pull-down menus. The use of R functions becomes a simple cut-n-paste activity, supplying the required information in the argument statements. There are several Internet web sites that offer information, resources, and assis- tance with R, R programs, and examples. These can be located by entering “R soft- ware” in the search engines accessible from any Internet browser software. The main Internet URL (Uniform Resource Locator) address for R is: h ttp://www.r- project.org . A second URL is: http://lib.stat.cmu.edu/R/CRAN . There are also many websites offering R information, statistics, and graphing, for example, Quick-R at http://www.statmethods.net . Tuscaloosa, AL, USA Randall Schumacker Sara Tomek