Applied Statistics for Social and Management Sciences

447 Pages·4.598 MB·English

by  Miah A.Q.

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
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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