Abdul Quader Miah Applied Statistics for Social and Management Sciences Applied Statistics for Social and Management Sciences Abdul Quader Miah Applied Statistics for Social and Management Sciences 123 AbdulQuader Miah AsianInstitute of Technology Bangkok Thailand ISBN978-981-10-0399-8 ISBN978-981-10-0401-8 (eBook) DOI 10.1007/978-981-10-0401-8 LibraryofCongressControlNumber:2015960235 ©SpringerScience+BusinessMediaSingapore2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. When you can measure what you are speaking about, and express it in numbers, you know something about it. When you cannot measure it, when you cannot express it in numbers, yourknowledge is of ameagerand unsatisfactory kind. It may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the stage of science. Lord Kelvin Preface The beauty of statistics lies in the fact that almost all planning, management, business and engineering problems, and phenomena require statistics as an ana- lytical tool to help in subsequent decision making. The users of these and other sciencesneedstatistics equally. To most of the users, the applied parts and notthe finemathematicsofthesubjectareofgreatimportance.Theyhavelittleopportunity and time in going to the sophisticated theories of statistics. At the same time, they must be in a position to interpret the technical details of the subject in order to correctly use the tool. This is important to guard against misuse of statistics. It is recognized that the users of statistics come largely from various disciplines, some having no strong mathematical background. Emphasis of such users is on the application of statistics in the real and practical problems. Myengineeringbackground,traininginplanningandmanagement,andteaching of statistics in the Asian Institute of Technology, Bangkok, have helped me in identifyingtheneedofthesubjectinawidevarietyofapplicationsinthereal-world perspectiveandtheproblemsthestudentsfaceinhandlingthestatisticaltechniques. Toovercomethese,alotoftechniquesandideaswereusedandfoundsuccessfulin the classroom. The purpose of writing this text book is to transfer these practical aspects into a reference book from which a wider range of readers can benefit. While I started teaching statistics to the students coming from various disciplines, the students were found fearful of the subject. My constant efforts were, therefore, to make the subject interesting to them rather than a fearful one. This book will reflect to some extent these efforts. The coverage of the book is evident from the Table of Contents. The chapter “Index Numbers” with their construction techniques, applications, and interpreta- tions is not normally available in ordinary statistics books. Yet the planning and management students need it frequently and to them it is of a great help. This has been added for their benefit. Bangkok Abdul Quader Miah April 2015 vii Contents 1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Contents of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Level of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Presentation of Statistical Data. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Tabular Presentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Graphical Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.1 Bar Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.3 Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.4 Frequency Polygon. . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.5 Pareto Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.6 Line Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.7 Frequency Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Descriptive Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Central Tendency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.3 Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.4 Comparison of Mean, Median, and Mode . . . . . . . . . 42 3.2 Measures of Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Interquartile Range (IQR) . . . . . . . . . . . . . . . . . . . . 44 3.2.3 Mean Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.4 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ix x Contents 3.2.5 Standard Deviation. . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.6 Coefficient of Variation. . . . . . . . . . . . . . . . . . . . . . 51 3.2.7 Stem-and-Leaf Diagram. . . . . . . . . . . . . . . . . . . . . . 51 3.2.8 Other Measures of Dispersion . . . . . . . . . . . . . . . . . 54 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Probability Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Probability Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Two Approaches in Calculating Probability. . . . . . . . . . . . . . . 60 4.3 Axioms of Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Probability in Mutually Exclusive Events . . . . . . . . . . . . . . . . 61 4.5 Probability in Independent Events . . . . . . . . . . . . . . . . . . . . . 62 4.6 Probability in Dependent Events (Conditional/Unconditional Probability). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.7 Probability in Non-mutually Exclusive Events . . . . . . . . . . . . . 63 4.8 Probability and Number of Possible Samples. . . . . . . . . . . . . . 64 4.8.1 Sampling with Replacement. . . . . . . . . . . . . . . . . . . 64 4.8.2 Sampling Without Replacement (Order Important) . . . 64 4.8.3 Sampling Without Replacement (Order Irrelevant) . . . 65 5 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 Discrete Probability Distribution. . . . . . . . . . . . . . . . . . . . . . . 70 5.1.1 The Binomial Distribution. . . . . . . . . . . . . . . . . . . . 70 5.1.2 Multinomial Probability Distribution. . . . . . . . . . . . . 72 5.1.3 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . 74 5.1.4 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.5 Important Features . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Continuous Probability Distribution . . . . . . . . . . . . . . . . . . . . 78 5.3 The Normal Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.1 Properties/Characteristics of Normal Distribution . . . . 79 5.3.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 The t Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.5 The F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6 The Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.7 Joint Probability Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 89 5.7.1 Discrete Joint Probability Distribution. . . . . . . . . . . . 89 5.7.2 Continuous Joint Probability Distribution. . . . . . . . . . 90 5.8 Data Fitting to Probability Distribution. . . . . . . . . . . . . . . . . . 91 6 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 Parameter and Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 Properties of Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3.1 Unbiasedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Contents xi 6.3.3 Sufficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3.4 Consistency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.5 Some Examples in Estimation . . . . . . . . . . . . . . . . . . . . . . . . 105 6.6 Point Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.7 Interval Estimation/Confidence Interval of the Mean of a Single Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.8 Confidence Interval of the Difference of Means of Two Normal Populations. . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.9 Confidence Interval of the Variance of a Normal Population . . . 112 6.10 Confidence Interval of a Population Proportion . . . . . . . . . . . . 113 6.11 Confidence Interval of the Difference of Two Population Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.12 Finite Population Correction Factor . . . . . . . . . . . . . . . . . . . . 115 7 Hypothesis Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Test Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.3 Hypothesis Testing—One Population Mean (Variance Known) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.3.1 One-Tail Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.3.2 Two-Tail Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.4 Hypothesis Testing—One Population Mean (Variance Unknown—Large Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.5 Hypothesis Testing—Equality of Two Population Means (Variance Known) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.5.1 One-Tail Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.5.2 Two-Tail Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.6 Hypothesis Testing—One Population Mean (Variance Unknown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.6.1 One-Tail Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.6.2 Two-Tail Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.7 Hypothesis Testing—Equality of Two Population Means (Variance Unknown—Small Sample) . . . . . . . . . . . . . . . . . . . 144 7.7.1 Situation 1: r2 ¼r2 ¼r2 . . . . . . . . . . . . . . . . . . . . 145 1 2 7.7.2 Situation 2: r2 6¼r2 . . . . . . . . . . . . . . . . . . . . . . . . 147 1 2 7.8 Testing of Hypothesis—Population Proportion. . . . . . . . . . . . . 149 7.8.1 One Population Proportion. . . . . . . . . . . . . . . . . . . . 150 7.8.2 Equality of Two Population Proportions . . . . . . . . . . 153 7.9 Power of Hypothesis Testing. . . . . . . . . . . . . . . . . . . . . . . . . 157 8 The Chi-Square Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.1 Goodness-of-Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.2 Test of Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 xii Contents 9 Nonparametric Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.1 The Sign Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.2 The Rank Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 9.2.1 The Wilcoxon Rank-Sum Test. . . . . . . . . . . . . . . . . 190 9.2.2 The Spearman Rank Correlation. . . . . . . . . . . . . . . . 194 9.3 Nonparametric Method in Analysis of Variance. . . . . . . . . . . . 196 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11 Simple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.1 Simple Linear Regression Model . . . . . . . . . . . . . . . . . . . . . . 215 11.2 Hypothesis Testing in Simple Linear Regression . . . . . . . . . . . 218 11.3 Confidence Intervals of Parameters. . . . . . . . . . . . . . . . . . . . . 220 11.4 Adequacy of the Regression Model . . . . . . . . . . . . . . . . . . . . 221 11.4.1 Residual Analysis to Test Uncorrelated Errors . . . . . . 221 11.4.2 Residual Analysis to Test Normality. . . . . . . . . . . . . 222 11.4.3 Lack-of-Fit Test. . . . . . . . . . . . . . . . . . . . . . . . . . . 222 11.5 Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . . . . . 224 11.6 Data Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 11.7 Interpretation of Simple Regression Model . . . . . . . . . . . . . . . 227 12 Multiple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 12.1 Multiple Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . 233 12.2 Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 12.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 12.4 Use of Dummy Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 12.5 Other Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 13 Sampling Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 13.1 Advantages of Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 13.2 Considerations Prior to Sample Survey. . . . . . . . . . . . . . . . . . 246 13.3 Considerations in Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . 247 13.4 Principal Steps Involved in the Choice of a Sample Size. . . . . . 248 13.5 Types of Commonly Used Sampling Methods. . . . . . . . . . . . . 249 13.5.1 Simple Random Sampling. . . . . . . . . . . . . . . . . . . . 249 13.5.2 Systematic Sampling. . . . . . . . . . . . . . . . . . . . . . . . 253 13.5.3 Cluster Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 254 13.5.4 Stratified Random Sampling. . . . . . . . . . . . . . . . . . . 255 14 Determination of Sample Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 14.1 Basic Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 14.2 Sample Size in the Case of Random Sampling (Continuous Data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 14.3 Sample Size in Case of Simple Random Sampling (Proportion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260