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SCHAUM’S OUTLINE OF THEORY AND PROBLEMS OF BEGINNING STATISTICS LARRY J. STEPHENS, Ph.D. Professor of Mathematics University of Nebrasku at Oriialin SCHAUM’S OUTLINE SERIES McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogotci Caracas Lisbon London Madrid Mexico City Milan Montreal New Dehli San Juan Singapore Sydney Tokyo Toronto To My Mother and Father, Rosie, and Johnie Stephens LARRY J. STEPHENS is Professor of Mathematics at the University of Nebraska at Omaha. He received his bachelor’s degree from Memphis State University in Mathematics, his master’s degree from the University of Arizona in Mathematics, and his Ph.D. degree from Oklahoma State University in Statistics. Professor Stephens has over 40 publications in professional journals. He has over 25 years of experience teaching Statistics. He has taught at the University of Arizona, Christian Brothers College, Gonzaga University, Oklahoma State University, the University of Nebraska at Kearney, and the University of Nebraska at Omaha. He has published numerous computerized test banks to accompany elementary Statistics texts. He has worked for NASA, Livermore Radiation Laboratory, and Los Alamos Laboratory. Since 1989, Dr. Stephens has consulted with and conducted Statistics seminars for the engineering group at 3M, Valley, Nebraska plant. Schaurn’s Outline of Theory and Problems of BEGlNNlNG STATISTICS Copyright 0 1998 by the McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of Atnerica. Except as permitted under the Copyright Act of 1976, no part of this publication tna he reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 3 4 5 6 7 8 9 10 I1 12 13 14 IS 16 17 18 19 20 PRS PRS 9 0 2 1 0 9 ISBN 0-07-06 1259-5 Sponsoring Editor: Barbara Gilson Production Supervisor: Clara Stanley Editing Supervisor: Maureen B. Walker Mirrirtih is U registered tradenuirk of Minitub hc. Library of Congress Cataloging-in-PublicationD ata Stephens. Larry J. Schaum’s outline of theory and problems of beginning statistics / Larry J. Stephens. p. cm. - (Schaurn’s outline series) Includes index. ISBN 0-07-06 1259-5 (pbk.) I. Mathematical statistics-Outlines, syllabi, etc. 2. Mathematical statistics-Problems, exercises, etc. I. Title. 11. Series. QA276.19.S74 1998 519.5’0764~21 97-45979 CIP AC Preface Statistics is a required course for undergraduate college students in a number of majors. Students in the following disciplines are often required to take a course in beginning statistics: allied health careers, biology, business, computer science, criminal justice, decision science, engineering, education, geography, geology, information science, nursing, nutrition, medicine, pharmacy, psychology, and public administration. This outline is intended to assist these students in the understanding of Statistics. The outline may be used as a supplement to textbooks used in these courses or a text for the course itself. The author has taught such courses for over 25 years and understands the difficulty students encounter with statistics. I have included examples from a wide variety of current areas of application in order to motivate an interest in learning statistics. As we leave the twentieth century and enter the twenty-first century, an understanding of statistics is essential in understanding new technology, world affairs, and the ever-expanding volume of knowledge. Statistical concepts are encountered in television and radio broadcasting, as well as in magazines and newspapers. Modern newspapers, such as USA Today, are full of statistical information. The sports section is filled with descriptive statistics concerning players and teams performance. The money section of USA Today contains descriptive statistics concerning stocks and mutual funds. The life section of USA Today often contains summaries of research studies in medicine. An understanding of statistics is helpful in evaluating these research summaries. The nature of the beginning statistics course has changed drastically in the past 30 or so years. This change is due to the technical advances in computing. Prior to the 1960s statistical computing was usually performed on mechanical calculators. These were large cumbersome computing devices (compared to today’s hand-held calculators) that performed arithmetic by moving mechanical parts. Computers and computer software were no comparison to today’s computers and software. The number of statistical packages available today numbers in the hundreds. The burden of statistical computing has been reduced to simply entering your data into a data file and then giving the correct command to perform the statistical method of interest. One of the most widely used statistical packages in academia as well as industrial settings is the package called Minitab (Minitab Inc., 3081 Enterprise Drive, State College, PA 16801-3008). I wish to thank Minitab Inc. for granting me permission to include Minitab output, including graphics, throughout the text. Most modern Statistics textbooks include computer software as part of the text. I have chosen to include Minitab because it is widely used and is very friendly. Once a student learns the various data file structures needed to use Minitab, and the structure of the commands and subcommands, this knowledge is readily transferable to other statistical software. The outline contains all the topics, and more, covered in a beginning statistics course. The only mathematical prerequisite needed for the material found in the outline is arithmetic and some basic algebra. I wish to thank my wife, Lana, for her understanding during the preparation of the book. I wish to thank my friend Stanley Wileman for all the computer help he has given me during the preparation of the book. I wish to thank Dr. Edwin C. Hackleman of Delta Software, Inc. for his timely assistance as compositor of the final camera-ready manuscript. Finally, I wish to thank the staff at McGraw-Hill for their cooperation and helpfulness. LARRYJ . STEPHENS ... 111 This page intentionally left blank Contents Chapter 1 INTRODUCTION ................................................................................. 1 Statistics. Descriptive Statistics. Inferential Statistics: Population and Sample. Variable, Observation. and Data Set. Quantitative Variable: Discrete and Continuous Variable. Qualitative Variable. Nominal, Ordinal, Interval, and Ratio Levels of Measurement. Summation Notation. Computers and Statistics. Chapter 2 ORGANIZING DATA .......................................................................... 14 Raw Data. Frequency Distribution for Qualitative Data. Relative Frequency of a Category. Percentage. Bar Graph. Pie Chart. Frequency Distribution for Quantitative Data. Class Limits, Class Boundaries, Class Marks, and Class Width. Single-Valued Classes. Histograms. Cumulative Frequency Distributions. Cumulative Relative Frequency Distributions. Ogives. Stem-and-Leaf Displays. Chapter 3 DESCRIPTIVE MEASURES .............................................................. 40 Measures of Central Tendency. Mean, Median, and Mode for Ungrouped Data. Measures of Dispersion. Range, Variance, and Standard Deviation for Ungrouped Data. Measures of Central Tendency and Dispersion for Grouped Data. Chebyshev’s Theorem. Empirical Rule. Coefficient of Variation. Z Scores. Measures of Position: Percentiles, Deciles, and Quartiles. Interquartile Range. Box-and-Whisker Plot. Chapter 4 PROBABILITY ..................................................................................... 63 Experiment, Outcomes, and Sample Space. Tree Diagrams and the Counting Rule. Events, Simple Events, and Compound Events. Probability. Classical, Relative Frequency and Subjective Probability Definitions. Marginal and Conditional Probabilities. Mutually Exclusive Events. Dependent and Independent Events. Complementary Events. Multiplication Rule for the Intersection of Events. Addition Rule for the Union of Events. Bayes’ Theorem. Permutations and Combinations. Using Permutations and Combinations to Solve Probability Problems. Chapter 5 DISCRETE RANDOM VARIABLES ................................................ 89 Random Variable. Discrete Random Variable. Continuous Random Variable. Probability Distribution. Mean of a Discrete Random Variable. Standard Deviation of a Discrete Random Variable. Binomial Random Variable. Binomial Probability Formula. Tables of the Binomial Distribution. Mean and Standard Deviation of a Binomial Random Variable. Poisson Random Variable. Poisson Probability Formula. Hypergeome tric Random Variable. Hypergeometric Probability Formula. V vi CONTENTS Chapter 6 CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS. ................................................... 113 Uniform Probability Distribution. Mean and Standard Deviation for the Uniform Probability Distribution. Normal Probability Distribution. Standard Normal Distribution. Standardizing a Normal Distribution. Applications of the Normal Distribution. Determining the z and x Values When an Area under the Normal Curve is Known. Normal Approximation to the Binomial Distribution. Exponential Probability Distribution. Probabilities for the Exponential Probability Distribution. Chapter 7 SAMPLING DISTRIBUTIONS .......................................................... 140 Simple Random Sampling. Using Random Number Tables. Using the Computer to Obtain a Simple Random Sample. Systematic Random Sampling. Cluster Sampling. Stratified Sampling. Sampling Distribution of the Sampling Mean. Sampling Error. Mean and Standard Deviation of the Sample Mean. Shape of the Sampling Distribution of the Sample Mean and the Central Limit Theorem. Applications of the Sampling Distribution of the Sample Mean. Sampling Distribution of the Sample Proportion. Mean and Standard Deviation of the Sample Proportion. Shape of the Sampling Distribution of the Sample Proportion and the Central Limit Theorem. Applications of the Sampling Distribution of the Sample Proportion. Chapter 8 ESTIMATION AND SAMPLE SIZE DETERMINATION: ONE POPULATION ........................................................................... 166 Point Estimate. Interval Estimate. Confidence Interval for the Population Mean: Large Samples. Maximum Error of Estimate for the Population Mean. The t Distribution. Confidence Interval for the Population Mean: Small Samples. Confidence Interval for the Population Proportion: Large Samples. Determining the Sample Size for the Estimation of the Population Mean. Determining the Sample Size for the Estimation of the Population Proportion. Chapter 9 TESTS OF HYPOTHESIS: ONE POPULATION ............................ 185 Null Hypothesis and Alternative Hypothesis. Test Statistic, Critical Values, Rejection and Nonrejection Regions.Type I and Type I1 Errors. Hypothesis Tests about a Population Mean: Large Samples. Calculating Type I1 Errors. P Values. Hypothesis Tests about a Population Mean: Small Samples. Hypothesis Tests about a Population Proportion: Large Samples. CONTENTS vii Chapter 10 INFERENCES FOR TWO POPULATIONS ..................................... 211 x, Sampling Distribution of - for Large Independent Samples. Estimation of p1- p2 Using Large Independent Samples. Testing Hypothesis about pI- p, Using Large Independent Samples. -x2 Sampling Distribution of XI for Small Independent Samples from Normal Populations with Equal (but unknown) Standard Deviations. Estimation of p1- p, Using Small Independent Samples froin Normal Populations with Equal (but unknown) Standard Deviations. Testing Hypothesis about p, - p, Using Small Independent Samples from Normal Populations with Equal (but Unknown) Standard x2 Deviations. Sampling Distribution of XI- for Small Independent Samples from Normal Populations with Unequal (and Unknown) Standard Deviations. Estimation of p, - p2 Using Small Independent Samples from Normal Populations with Unequal (and Unknown) Standard Deviations. Testing Hypothesis about p1- p2 Using Small Independent Samples from Normal Populations with Unequal (and Unknown) Standard Deviations. a Sampling Distribution of for Normally Distributed Differences Computed from Dependent Samples. Estimation of pd Using Normally Distributed Differences Computed from Dependent Samples. Testing Hypothesis about pd Using Normally Distributed Differences Computed from Dependent Samples. Sampling Distribution of PI- p2 for Large Independent Samples. Estimation of PI- P2 Using Large Independent Samples. Testing Hypothesis about PI- P2 Using Large Independent Samplcs. Chapter 11 CHI-SQUARE PROCEDURES. .......................................................... 249 Chi-square Distribution. Chi-square Tables. Goodness-of-Fit Test. Observed and Expected Frequencies. Sampling Distribution of the Goodness-of-Fit Test Statistic. Chi-square Independence Test. Sampling Distribution of the Test Statistic for the Chi-square Independence Test. Sampling Distribution of the Sample Variance. Inferences Concerning the Population Variance. Chapter 12 ANALYSIS OF VARIANCE (ANOVA) ............................................. 272 F Distribution. F Table. Logic Behind a One-way ANOVA. Sum of Squares, Mean Squares, and Degrees of Freedom for a One-way ANOVA. Sampling Distribution for the One-way ANOVA Test Statistic. Building One-way ANOVA Tables and Testing the Equality of Means. Logic Behind a Two-way ANOVA. Sum of Squares, Mean Squares, and Degrees of Freedom for a Two-way ANOVA. Building Two-way ANOVA Tables. Sampling Distributions for the Two-way ANOVA. Testing Hypothesis Concerning Main Effects and Interaction. ... -~ CONTENTS Vlll Chapter 13 REGRESSION AND CORRELATION. ............................................. 309 Straight Lines. Linear Regression Model. Least Squares Line. Error Sum of Squares. Standard Deviation of Errors. Total Sum of Squares. Regression Sum of Squares. Coefficient of Determination. hiean, Standard Deviaticjii, and Sampling Distribution of the Slope of the Estimated Regression Equation. Inferences Concerning the Slope of the Population Regression Line. Estimation and Prediction in Linear Regression. Linear Correlation Coefficient. Inference Concerning the Population Correlation Coefficient. Chapter 14 NONPARAMETRIC STATISTICS... ................................................. 334 NonparametricMethods. Sign Test. Wilcoxon Signed-Ranks Test for Two Dependent Samples. Wilcoxon Rank-Sum Test for Two Independent Samples. Kruskal-Wall6 Test. Rank Correlation. Runs Test for Randomness. APPENDIX 1 Binomial Probabilities ....................................................................................................................... 359 2 Areas under the Standard Normal Curve from 0 to 2 ....................................................................... 364 ~~- 3 Area in the Right Tail under the t Distribution Curve ....................................................................... 365 ~ ... 4 Area in the Right Tail under the Chi-square Distribution Curve ....................................................... 366 ~ 5 Area in the Right Tail under the F Distribution Curve ...................................................................... 367 ~ INDEX .................................................................................................................................................... 369 - Chapter 1 Introduction STATISTICS Statistics is a discipline of study dealing with the collection, analysis, interpretation, and presentation of data. Statistical methodology is utilized by pollsters who sample our opinions concerning topics ranging from art to zoology. Statistical methodology is also utilized by business and industry to help control the quality of goods and services that they produce. Social scientists and psychologists use statistical methodology to study our behaviors. Because of its broad range of applicability, a course in statistics is required of majors in disciplines such as sociology, psychology, criminal justice, nursing, exercise science, pharmacy, education, and many others. To accommodate this diverse group of users, examples and problems in this outline are chosen from many different sources. DESCRIPTIVE STATISTICS The use of graphs, charts, and tables and the calculation of various statistical measures to organize and summarize information is called descriptive statistics. Descriptive statistics help to reduce our information to a manageable size and put it into focus. EXAMPLE 1.1 The compilation of batting average, runs batted in, runs scored, and number of home runs for each player, as well as earned run average, wordlost percentage, number of saves, etc., for each pitcher from the official score sheets for major league baseball players is an example of descriptive statistics. These statistical measures allow us to compare players, determine whether a player is having an “off year” or “good year,” etc. EXAMPLE 1.2 The publication entitled Crime in the United States published by the Federal Bureau of Investigation gives summary information concerning various crimes for the United States. The statistical measures given in this publication are also examples of descriptive statistics and they are useful to individuals in law enforcement. INFERENTIAL STATISTICS: POPULATION AND SAMPLE The complete collection of individuals, items, or data under consideration in a statistical study is referred to as the pupulatiurz. The portion of the population selected for analysis is called the sample. Inferential statistics consists of techniques for reaching conclusions about a population based upon information contained in a sample. EXAMPLE 1.3 The results of polls are widely reported by both the written and the electronic media. The techniques of inferential statistics are widely utilized by pollsters. Table 1.1 gives several examples of populations and samples encountered in polls reported by the media. The methods of inferential statistics are used to make inferences about the populations based upon the results found in the samples and to give an indication about the reliability of these inferences. The results of a poll of 600 registered voters might be reported as follows: Forty percent of the voters approve of the president’s economic policies. The margin of error for the survey is 4%. The survey indicates that an estimated 40% of all registered voters approve of the economic policies, but it might be as low as 36% or as high as 44%. 1