Statistics for Business and Economics: Compendium of Essential Formulas, Third Edition
Franz W. Peren
Contents
About the Author XV
List of Abbreviations XVII
1 Statistical Signs and Symbols 1
2 Descriptive Statistics 3
2.1 Empirical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Cumulative Frequencies . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Mean Values and Measures of Dispersion . . . . . . . . . . . . 6
2.2.1 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Ratios and Index Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Index Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 Peren-Clement Index (PCI) . . . . . . . . . . . . . . . . . . . . 38
2.4 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.1 Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . 51
2.5.1.1 Confidence Intervals for the Regression
Coefficients of a Simple Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.1.2 Student’s t-Tests for the Regression Coefficients
of a Simple Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.2 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . 62
2.5.2.1 Confidence Intervals for the Regression
Coefficients of a Multiple Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5.2.2 Student’s t-Tests for the Regression Coefficients
of a Multiple Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5.3 Double Linear Regression . . . . . . . . . . . . . . . . . . . . 66
2.5.3.1 Confidence Intervals for the Regression
Coefficients of a Double Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.3.2 Student’s t-Tests for the Regression Coefficients
of a Double Linear Regression
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.4 Dummy Variables in Regression Analysis . . . . . . . 76
2.5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.4.2.1 Derivation of the Simple and
Multiple Regression Function . . 76
2.5.4.2.2 Estimation of the Regression
Parameters according to the
Least Squares Method . . . . . . . 77
2.5.4.2.3 Integration of Categorial Influences
via Dummy Variables . . . 79
2.5.4.3 Quality Measures . . . . . . . . . . . . . . . . . . . . . 82
2.5.4.3.1 Goodness of Fit (Global Quality
Criteria) . . . . . . . . . . . . . . . . . . 82
2.5.4.3.2 Quality Criteria of the Regression
Coefficients . . . . . . . . . . . . . 85
2.5.4.3.3 Checking the Model Assumptions. . . . . . . . . . . . . . . . . . . . . . . 87
2.5.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.5.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3 Inferential Statistics 99
3.1 Probability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1.1 Fundamental Terms/Definitions . . . . . . . . . . . . . . . . 99
3.1.2 Theorems of Probability Theory. . . . . . . . . . . . . . . . 104
3.2 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.1 Concept of Random Variables . . . . . . . . . . . . . . . . . 110
3.2.2 Probability, Distribution and Density Function . . . . 111
3.2.2.1 Discrete Random Variables . . . . . . . . . . . . 111
3.2.2.2 Continuous Random Variables . . . . . . . . . 113
3.2.3 Parameters for Probability Distributions . . . . . . . . . 116
3.3 Theoretical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3.1 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3.2 Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . 128
3.4 Statistical Estimation Methods (Confidence Intervals) . . 148
3.4.1 Confidence Interval for the Arithmetic Mean of
the Population μ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.4.2 Confidence Interval for the Variance of the
Population σ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.4.3 Confidence Interval for the Share Value in the
Population θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.4.4 Confidence Interval for the Difference of the Mean
Values of Two Populations μ1 and μ2 . . . . . . . . . . . 156
3.4.5 Conficence Interval for the Difference of the Share
Values of Two Populations θ1 and θ2 . . . . . . . . . . . . 160
3.5 Determination of the Required Sample Size . . . . . . . . . . . 163
3.5.1 Determination of the Required Sample Size for
an Estimation of the Arithmetic Mean μ . . . . . . . . . 163
3.5.2 Determination of the Required Sample Size for
an Estimation of the Share Value θ . . . . . . . . . . . . . 165
3.6 Statistical Testing Methods . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.6.1 Parameter Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3.6.1.1 Arithmetic Mean with Known Variance
of the Population | One Sample Test . . . . . 167
3.6.1.2 Arithmetic Mean with Unknown Variance
of the
Population | One Sample Test . . . . . . . . . . 170
3.6.1.3 Share Value | One Sample Test . . . . . . . . . 172
3.6.1.4 Variance | One Sample Test . . . . . . . . . . . . 174
3.6.1.5 Difference of Two Arithmetic Means with
Known Variances of the Population | Two
Samples Test . . . . . . . . . . . . . . . . . . . . . . . . 177
3.6.1.6 Difference of Two Arithmetic Means with
Unknown Variances of the Populations
under the Assumption that their Variances
are Unequal | Two Samples Test . . 179
3.6.1.7 Difference of Two Arithmetic Means with
Unknown Variances of the Populations
under the Assumption that their Variances
are Equal | Two Samples Test . . . . 182
3.6.1.8 Difference of Two Share Values | Two
Samples Test . . . . . . . . . . . . . . . . . . . . . . . . 185
3.6.1.9 Quotients of Two Variances | Two Samples
Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
3.6.2 Distribution Tests (Chi-Squared Tests) . . . . . . . . . . 191
3.6.2.1 Chi-Squared Goodness of Fit Test . . . . . . 191
3.6.2.1.1 Chi-Squared Goodness of Fit
Test for a Discrete Distribution
of the Population . . . . . . . . . . . . . 191
3.6.2.1.2 Chi-Squared Goodness of Fit
Test for a Continous Distribution
of the Population . . . . . . . . . 197
3.6.2.2 Chi-Squared Independence Test . . . . . . . . 201
3.6.2.3 Chi-Squared Homogeneity Test . . . . . . . . . 207
3.6.3 Yates’s Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4 Probability Calculation 215
4.1 Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.2 Definitions of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.2.1 The Classical Definition of Probabilty . . . . . . . . . . . 216
4.2.2 The Statistical Definition of Probability . . . . . . . . . . 217
4.2.3 The Subjective Definition of Probability . . . . . . . . . 218
4.2.4 Axioms of Probability Calculation . . . . . . . . . . . . . . 218
4.3 Theorems of Probability Calculation . . . . . . . . . . . . . . . . . 220
4.3.1 Theorem of Complementary Events . . . . . . . . . . . . 220
4.3.2 The Multiplication Theorem with Independence
of Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.3.3 The Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . 221
4.3.4 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . 223
4.3.5 Stochastic Independence . . . . . . . . . . . . . . . . . . . . . 224
4.3.6 The Multiplication Theorem in General Form . . . . 225
4.3.7 The Theorem of Total Probability . . . . . . . . . . . . . . . 225
4.3.8 Bayes’ Theorem (Bayes’ Rule) . . . . . . . . . . . . . . . . . 227
4.3.9 Overview of the Probability Calculation of Mutually
Exclusive and Non-Exclusive Events . . . . . . . . 230
4.4 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.4.1 The Concept of Random Variables . . . . . . . . . . . . . 231
4.4.2 The Probability Function of Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.4.3 The Distribution Function of Discrete Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
4.4.4 Probability Density and Distribution Function of
Continuous Random Variables . . . . . . . . . . . . . . . . . 234
4.4.5 Expected Value and Variance of Random Variables 239
A Statistical Tables 243
B Case Study: Regression Analysis Using Dummy
Variables in R 321
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
B.2 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
B.3 Conducting the Regression Analysis . . . . . . . . . . . . . . . . . 325
B.3.1 Single Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
B.3.2 Multiple Regression with Dummy Variables . . . . . . 327
C Bibliography 331
Index 341