Dr. Manish SharmaDirector GeneralRBMI Group of Institutions,Bareilly and Greater Noida

Amit GuptaAsst. ProfessorDeterment of CS & ITRakshpal Bahadur Management Institute, Bareilly

The Practice of

B U S I N E S S S T A T I S T I C S

Using Statistics For Decision Making

List of Reviewers

The authors and the publishers gratefully acknowledge all the reviewers for their praise as well as constructive analysis of the book. Here is a brief list.

Dr. R. P. Saxena Punjab Technical University, Punjab

Dr. Manish Goel ABS Management Institute, Bharuch

Vikas Gupta Regional HR Head (North) GKB Lens Rx, New Delhi

Prasant Gupta FIT, Meerut

Preface

It gives us great pleasure in presenting the first edition of the book “The Practice of Business Statistics” to our esteemed readers; it is a unique book valuable for beginners, intermediate and advance users.

Today’s good decisions are driven by data. In all aspects of our lives, and more so in the business context, an amazing diversity of data is available for inspection and analytical insight. Business managers and professionals are increasingly required to justify decisions on the basis of data. They need statistical model-based decision support systems.

Statistical skills enable them to intelligently collect, analyze and interpret data relevant to their decision-making. Statistical concepts and statistical thinking enable them to: • Solve problems in a diversity of contexts. • Add substance to decisions. • Reduce guesswork.

Business Statistics is a science assisting you to make business decisions under uncertainties based on some numerical and measurable scales. Decision making processes must be based on data, neither on personal opinion nor on belief.

This book has been written as a practical response to the needs of students who want to obtain a reasonable grasp of basic statistical techniques or methods in a limited time. The emphasis throughout the book is on understanding through practice. This will encourage students who lack confidence in their mathematical ability to understand statistical techniques.

An Invaluable and Ideal text for

Undergraduate and postgraduate students of Management at various levels as well as reference for professional students. Other than providing useful guidance to the students in various professional and competitive examinations, this book should serve as core textbook for the students of: • MBA, PGDBM, M.Com., MA(Eco) • MCA, BE, B-Tech. (CS & IT) • CA, ICWA,AMIE • BBA, BCA, B. Com

x The Practice of Business Statistics

Distinguishing Features

Some of the most distinguishing features are: • Each chapter provides a step-wise development of the subject matter under reference. • More than 800 solved examples. (With CD), on occasions, deliberately very simple examples are

given to highlight the theoretical aspects of statistics so that the concepts do not get submerged in complicated numerical examples.

• Each chapters ends with “List of Formulae” which are useful for a quick review of the subject. • Special emphasis has been given on the topics like Probability & Probability Distributions to

make them as simple as possible to be within the understanding of the students. • A detailed discussion of simple regression & correlation analysis. In simple regression (Chapter

11), we give concise explanations of the simple linear regression model, least squares, and confidence and prediction intervals.

• A simple and easy to understand example introducing sampling distributions. • A step-by-step Hypothesis-Testing approach that is used in almost all hypothesis-testing

examples in Chapter 16 (Hypothesis Testing). This approach consists of a five-step procedure that is designed to break hypothesis testing down into small, easy to understand steps and also to clearly show how to use the book’s hypothesis-testing summary boxes.

• Confidence intervals for and hypothesis tests about a population mean presented by using the known versus s unknown approach. This approach simplifies the choice of Z- or t-based procedures.

• Apart from simple language and lucid presentation, the emphasis is also on making the subject interesting by including interesting examples from day to day life and work environment.

With these features in view, we are sure that the book will be found useful by the students and teachers as well for whom it is primarily written.

Inspite of all efforts, some errors might be there, we shall be grateful to our readers if the same are brought to our notice. Suggestions and comments for further improvement of the book will be gratefully acknowledged.

Manish SharmaAmit Gupta

Acknowledgement

First and foremost, we would like to thank the Almighty God who gave us inspiration to take up this task. We would like to acknowledge our deep sense of gratitude to Mrs. Veena Mathur, Founder and

Chairperson of Rakshpal Bahadur Group of Institutes, Bareilly & Gr. Noida and to Er. Naveen Mathur, Managing Director of RBMI Group for their benevolent support, suggestions and guidance.

We would like to acknowledge our sincere thanks to our learned professors and authors of a number of excellent books on the subject, and inquisitive students who motivated us to understand the subject better and present it in an engaging manner.

Most important, we want to thank you, the reader, for considering the thoughts, tools, and techniques we lay out in these pages. We understand the importance of your business analyses, and appreciate that you are taking time to consider our ideas for using statistics to make better decisions on the potential, quality, and performance of your enterprise and your career. We hope this book, with its practical examples, will revolutionize your understanding of statistical analysis, without burdening you with the details that keep you from getting the business information you really need to know.

Mr. Punit Khanna and Mr. Mukul Seth, the publisher of this book, deserve special appreciation for the pain they have taken in bringing out this book in such an excellent form.

Finally, thanks to everyone at Khanna Publication who has played a role in getting this book onto the shelves.

Authors

How This Book is Organized

Welcome to “The Practice of Business Statistics”. This book Business Statistics is an outcome of long teaching experience of the subject. It includes the fundamental concepts, illustrative examples and application to various business problems. These illustrated examples have been selected carefully on each topic and sufficient numbers of unsolved questions are provided which aim at sharpening the skill of the students.

The chapters of the book could be classified to include conventional as well as unconventional statistical topics as follows: • Introduction-Meaning, Definition and Scope (Chapter-01) • Collection of Data, Classification and Tabulation (Chapter-02 and 03) • Presentation of Data: Diagrammatic & Graphic (Chapter-04 and 05) • Measures of Central Tendency, Dispersion, Skewness, Moments & Kurtosis (Chapter- 06,

07 and 08) • Index Numbers (Chapter-09) • Correction & Regression Analysis (Chapter-10 and 11) • Time Series Analysis & Forecasting (Chapter-12) • Probability & Probability Distributions (Chapter-13 and 14) • Sampling Distributions and Estimation (Chapter-15) • Statistical Inference encompassing Confidence Intervals, Test of Hypothesis, Chi-Square,

ANOVA (Chapter-16, 17 and 18) • Association of Attributes (Chapter-19) • Statistical Decision Theory (Chapter-20) • Statistical Quality Control (Chapter-21)Appendix

Appendix A of the book provides the various tables.

xiv The Practice of Business Statistics

Request to Readers

We request the readers to send their valuable suggestions for any modification or addition by way of text, examples. These would be gratefully acknowledged in the edition.

Manish SharmaAmit Gupta

PART — 1

1. Introduction — Meaning, Definition and Scope 1–22 1.1 Introduction 1 1.2 What is Statistics? 2 1.3 Meaning of Statistics 2 1.4 Definitions of Statistics 3 1.5 Branch of Statistics 6 1.5.1 Statistical Methods 6 1.6 What is Business Statistics? 10 1.7 Nature of Statistics: Science or Art 11 1.8 Importance and Scope of Statistics 11 1.9 Functions of Statistics 14 1.10 Uses of Statistics 15 1.11 Distrust of Statistics 15 1.12 Limitations of Statistics 17 1.13 Statistics of Software 19 Self Practice Exercises 19

2. Collection of Data 23–47 2.1 Introduction 23 2.2 Nature of Data 24 2.3 Type of Data and Levels of Measurement 25 2.4 Categories of Data 26 2.5 Primary Data 26 2.6 Drafting a Questionnaire 29

Contents

xvi The Practice of Business Statistics

2.6.1 What can questionnaires measure? 29 2.6.2 Characteristics of a Good Questionnaire 30 2.6.3 When to use a questionnaire? 30 2.7 Secondary Data 43 2.7.1 Sources of Secondary Data 43 Self Practice Exercises 45

3. Organising Data: Classification and Tabulation 48–72 3.1 Introduction 48 3.2 Classification of Data 49 3.3 Types of Classification 49 3.4 Frequency Distribution 52 3.4.1 Formation of a Frequency Distribution: Ungrouped Data 52 3.4.2 Formation of a Frequency Distribution: Discrete Data 52 3.4.3 Formation of Frequency Distribution: Continuous Data 54 3.4.4 Types of Class Intervals 55 3.5 Formation of Grouped Frequency Table 57 3.6 Relative Frequency and Percentage Distribution 60 3.7 Cumulative Frequency Distribution (CFD) 62 3.8 Bivariate Frequency Distribution (BFD) 63 3.9 Tabulation of Data 65 3.9.1 Type of Tables 67 Self Practice Exercises 70

4. Presentation of Data: Diagrammatic 73–92 4.1 Introduction 73 4.2 Diagrams 73 4.2.1 Characteristics of Diagrams 74 4.2.2 Rules for Constructing Diagrams 74 4.2.3 Limitations of Diagrams 74 4.3 Types of Diagrams 75 4.4 One-Dimensional Diagrams 75 4.5 Two-Dimensional Diagrams 82 4.6 Three-Dimensional Diagrams 87 4.7 Pictograms and Cartograms 88 4.8 Choice of a Suitable Diagram 88 Self Practice Exercises 89

5. Presentation of Data: Graphic 93–123 5.1 Introduction 93 5.2 Graphs 93

Contents xvii

5.2.1 Construction of Graphs 94 5.2.2 General Rules for Drawing Graphs 95 5.3 Types of Graphs 96 5.4 Time series Graphs or Historigrams 97 5.5 Frequency Graphs 101 Miscellaneous Solved Examples 114 Self Practice Exercises 119

6. Measures of Central Tendency 124–189 6.1 Introduction 124 6.2 Measures of Central Tendency 125 6.2.1 Objects of Measures of Central Tendency 125 6.2.2 Characteristics for a Good Average 125 6.3 Various Measures of Central Tendency 126 6.4 Arithmetic Mean 126 6.4.1 Calculation of Arithmetic Mean- Ungrouped or Raw Data 127 6.4.2 Calculation of Arithmetic Mean-Discrete Series 128 6.4.3 Calculation of Arithmetic Mean-Continuous Series 129 6.4.4 Charlier’s Accuracy Check 131 6.4.5 Properties of Arithmetic Mean 131 6.4.6 Some Special Types of Problems in Arithmetic Mean and their Solutions 134 6.4.7 Merits and Demerits of Arithmetic Mean 137 6.4.8 Limitations of Arithmetic Mean 138 6.4.9 Weighted Arithmetic Mean 138 6.5 Median 141 6.5.1 Calculation of Median-Ungrouped or Raw Data 142 6.5.2 Calculation of Median-Discrete Series 144 6.5.3 Calculation of Median — Continuous Series 145 6.5.4 Graphic Method for Location of Median 148 6.5.5 Merits and Demerits of Median 150 6.5.6 Uses of the Median 151 6.5.7 Measurements depending upon Median 151 6.6 Quartiles 151 6.6.1 Calculation of Quartiles-Ungrouped Data or Raw Data 152 6.6.2 Calculation of Quartiles-Discrete Series 152 6.6.3 Calculation of Quartiles-Continuous Series 153 6.7 Deciles 155 6.7.1 Calculation of Deciles-Ungrouped Data or Raw Data 155 6.7.2 Calculation of Deciles- Continuous Data 155 6.8 Percentiles 156 6.8.1 Calculation of Percentile-Ungrouped Data or Raw Data 157

xviii The Practice of Business Statistics

6.8.2 Calculation of Percentile-Continuous Data 157 6.9 Mode 158 6.9.1 Calculation of Mode- Ungrouped or Raw Data 159 6.9.2 Calculation of Mode- Discrete and Continuous Distribution 160 6.9.3 Determination of Modal Class 161 6.9.4 Graphic Location of Mode 163 6.9.5 Merits and Demerits of Mode 164 6.9.6 Uses of the Mode 164 6.10 Empirical Relationship Among Averages 164 6.11 Comparison of the Mean, Median and Mode 166 6.12 Selecting Among the Mode, Median and Mean 167 6.13 Difference among Mode, Median and Mean 168 6.14 Which of The Three Measures is The Best? 168 6.15 Harmonic Mean (H.M) 169 6.15.1 Merits and Demerits of Harmonic Mean 171 6.16 Geometric Mean (G.M) 172 6.16.1 Merits and Demerits of Geometric Mean 174 6.17 Defining the Arithmetic, Geometric and Harmonic Means 174 6.18 Relationship among A.M, G.M and H.M 175 Miscellaneous Solved Examples 176 List of Formulae 181 Self Practice Exercises 184

7. Measures of Dispersion (Variation) 190–252 7.1 Introduction 190 7.2 Measures of Dispersion 190 7.2.1 Definition of Dispersion 191 7.2.2 Objects of Measuring Dispersion or Variability 192 7.2.3 Characteristics for an Ideal Measure of Dispersion 192 7.3 Methods for Measures of Dispersion 192 7.4 Range and Coefficient of Range 193 7.4.1 Calculation of Range — Individual series 194 7.4.2 Calculation of Range — Discrete Series 195 7.4.3 Calculation of Range — Continuous Series 195 7.4.4 Quartile and Interquartile Range 196 7.4.5 Percentile and Percentile Range 197 7.4.6 Merits and Demerits of Range 197 7.4.7 Uses of Range 198 7.5 Quartile Deviation and Coefficient of Quartile Deviation 198 7.5.1 Calculation of Quartile Deviation — Individual Series 199 7.5.2 Calculation of Quartile Deviation — Discrete Series 199

Contents xix

7.5.3 Calculation of Quartile Deviation — Continuous Series 200 7.5.4 Merits and Demerits of Quartile Deviation 202 7.6 Mean Deviation and Coefficient of Mean Deviation 202 7.6.1 Calculation of Mean Deviation — Ungrouped or Raw Data 203 7.6.2 Calculation of Mean Deviation — Discrete Series 208 7.6.3 Calculation of Mean Deviation — Continuous Series 209 7.6.4 Merits and Demerits of Mean Deviation 211 7.6.5 Uses of Mean Deviation 211 7.7 Standard Deviation (S.D) 211 7.7.1 Difference between Mean Deviation and Standard Deviation 212 7.7.2 Calculation of Standard Deviation: Ungrouped or Raw Data 212 7.7.3 Calculation of Standard Deviation — Discrete Series 218 7.7.4 Calculation of Standard Deviation — Continuous Series 219 7.7.5 Charlier’s Check of Accuracy 221 7.7.6 Calculation of Standard deviation from Summation Method 221 7.7.7 Mathematical Properties of Standard Deviation 222 7.7.8 Calculation of Missing items and Missing Frequencies 224 7.7.9 Calculation of Corrected Standard Deviation 226 7.7.10 Combined Standard Deviation 228 7.7.11 Setting the limits through Standard Deviation 229 7.7.12 Variance and Coefficient of Variance 231 7.7.13 Other Measures based upon Standard Deviation 237 7.7.14 Merits and Demerits of Standard Deviation 237 7.7.15 Uses of Standard Deviation 238 7.8 Relationship Between Measures of Dispersion 238 7.9 Selecting Among the Q.D, M.D and S.D 238 Miscellaneous Solved Examples 239 List of Formulae 247 Self Practice Exercises 249

8. Skewness, Moments and Kurtosis 253–279 8.1 Introduction 253 8.2 Skewness 253 8.2.1 Type of Skewness 254 8.2.2 Comparison between Symmetrical and Skewed Distribution 255 8.2.3 Test of Skewness 255 8.2.4 Difference between Dispersion and Skewness 256 8.3 Measures of Skewness 256 8.3.1 Karl-Pearson’s Coefficient of Skewness 257 8.3.2 Bowley’s Coefficient of Skewness 260 8.3.3 Kelly’s Coefficient of Skewness 263

xx The Practice of Business Statistics

8.3.4 Measure of Skewness based on Moments 265 8.4 Moments 265 8.4.1 Moment about Mean 266 8.4.2 Moment about Arbitrary Point 266 8.4.3 Relationship between Raw Moments and Central Moments 267 8.4.4 Coefficients of Skewness based on Moments 267 8.5 Kurtosis 269 8.5.1 Measure of Kurtosis 270 8.5.2 Difference between skewness and Kurtosis 271 Miscellaneous Solved Examples 271 List of Formulae 275 Self Practice Exercises 276

PART — 2

9. Index Numbers 280–328 9.1 Introduction 280 9.2 Definition of Index Numbers 280 9.3 Uses of Index Numbers 281 9.4 Characteristics of Index Numbers 281 9.5 Limitations of Index Numbers 282 9.6 Types of Index Numbers 282 9.7 Problems in the Construction of Index Numbers 283 9.8 Method of Construction of Index Numbers 284 9.9 Un-weighted Index Numbers 284 9.10 Weight Index Number 287 9.11 Quantity or Volume Index Number 294 9.12 Value Index Numbers 295 9.13 Tests of Consistency of Index Numbers 296 9.14 Chain Index Numbers 302 9.15 Base Shifting of Index Numbers 305 9.16 Splicing of Index Numbers 307 9.17 Deflating of Index Numbers 310 9.18 Consumer Price Index (Cost of Living Index) 312 Miscellaneous Solved Examples 316 List of Formulae 324 Self Practice Exercises 325

10. Correlation Analysis 329–265 10.1 Introduction 329 10.2 Meaning of Correlation 330

Contents xxi

10.3 Definition of Correlation 330 10.4 Uses of Correlation 330 10.5 Types of Correlation 330 10.6 Methods of Correlation 332 10.7 Karl-pearson’s Method 334 10.8 Properties of Correlation Coefficient 342 10.9 Correlation of Bivariate Grouped Data 343 10.10 Probable Error 346 10.11 Rank Correlation 347 10.11.1 Computation for Tied Observations 348 10.12 Coefficient of Concurrent Deviations 354 Miscellaneous Solved Examples 356 List Of Formulae 359 Self Practice Exercises 361

11. Regression Analysis 366–400 11.1 Introduction 366 11.2 Meaning of Regression 366 11.3 Uses of Regression Analysis 368 11.4 Difference between Correlation and Regression 368 11.5 Type of Regression 368 11.6 Linear Regression Equation 369 11.6.1 Regression Lines 369 11.6.2 Principle of ‘Least Squares’ 370 11.7 Methods of Regression Analysis 371 11.8 Graphic Method 371 11.9 Algebraic Methods 373 11.9.1 Regression Equations 373 11.9.2 Regression Coefficients 374 11.9.3 Deviations taken from Assumed Means 381 11.9.4 Regression Equation in Group Frequency Distribution 384 11.10 Properties of Regression Coefficient 385 11.11 Standard Error of An Estimate 388 11.12 Strengths and Limitations of The Regression 390 Miscellaneous Solved Examples 391 List of Formulae 394 Self Practice Exercises 397

12. Time Series Analysis and Forecasting 401–452 12.1 Introduction 401 12.2 What is the Importance of Time Series Analysis? 401

xxii The Practice of Business Statistics

12.3 Time Series Definitions 402 12.4 What are the Components of Time Series? 403 12.5 Analysis of Time Series (Models of Decomposition) 403 12.6 Preliminary Adjustments 404 12.7 What is Secular Trend? 405 12.8 What are the Methods for Measuring Trends? 406 12.9 What is Seasonal Variation? 420 12.10 What are the Methods of Measuring Seasonal Variation? 420 12.11 Deseasonalisation of data 428 12.12 What is Cyclical Variation? 430 12.13 What are the Methods for Measuring Cyclical Fluctuations? 430 12.14 What is Irregular Variation 432 12.15 What are the Methods for Measuring Irregular Variations? 432 12.16 Forecasting 432 12.17 Methods of Forecasting 434 Miscellaneous Solved Examples 440 List of Formulae 446 Self Practice Exercises 447

PART — 3

13. Probability and Mathematical Expectation 453–510 13.1 Introduction 453 13.2 Basic Concept 454 13.3 Definitions of Probability 458 13.4 Some Basic Concepts of Set Theory 460 13.5 How to assign probabilities? 463 13.6 General Computational Probability Rules 466 13.6.1 Addition Theorem on Probabilities 466 13.6.2 Multiplication Theorem on Probabilities 472 13.7 Conditional Probability 473 13.8 BAYES’ Theorem 474 13.9 Basic Rules of Probability 478 13.10 Principal of Permutation And Combination 483 Miscellaneous Solved Examples 486 13.11 Random Variable 497 13.12 Probability Mass Function (PMS) 498 13.13 Probability Density Function (PDF) 498 13.14 Mathematical Expectation 501 13.15 Moment Generating Function (MGF) 505 List of Formulae 506

Contents xxiii

Self Practice Exercises 506

14. Probability Theoretical Distributions 511–556 14.1 Introduction 511 14.2 Type of Theoretical Distributions 512 14.2.1 Discrete Probability Distributions 512 14.2.2 Continuous Probability Distributions 512 14.3 Binomial Distribution 513 14.4 Poisson Distribution 526 14.5 Normal Distribution 536 Miscellaneous Solved Examples 548 List of Formulae 552 Self Practice Exercises 553

PART — 4

15. Sampling, Sampling Distributions and Estimation 557–610 15.1 Introduction 557 15.2 Population 557 15.2.1 Finite Population and Infinite Population 557 15.3 Census Method 558 15.3.1 Merits and Limitations of Census Method 558 15.4 Sampling and Sample 558 15.4.1 Need for Sampling 558 15.4.2 Elements of Sampling Plan 559 15.4.3 Principles of Sampling 560 15.4.4 Sampling With and Without Replacement 560 15.4.5 Advantages and Limitation of Sampling 561 15.5 Population Parameters and Sample Statistics 561 15.6 Census Versus Sample Method 563 15.7 Types of Sampling 564 15.7.1 Difference Between Non-probability and Probability Sampling 564 15.8 Probability Sampling 564 15.8.1 Simple Random Sampling 565 15.8.2 Stratified Random Sampling 566 15.8.3 Systematic Sampling (Quasi-Random Sampling) 568 15.8.4 Cluster Sampling 570 15.9 Non-Probability Sampling 572 15.10 Choice of Sampling Techniques 573 15.11 Sampling and Non-sampling Errors 573

xxiv The Practice of Business Statistics

15.12 Sampling Distributions 575 15.12.1 Properties of Sampling Distribution 578 15.12.2 Sampling Distribution of Sample Mean 579 15.12.3 Sampling Distribution of Mean When Population has Non-normal Distribution 580 15.12.4 Central Limit Theorem (CLT) 580 15.12.5 Sampling Distribution of Mean When Population has Normal Distribution 582 15.12.6 Sampling Distribution of Difference Between Two Sample Means 586 15.12.7 Sampling Distribution of Sample Proportion 588 15.12.8 Sampling Distribution of the Difference of Two Proportions 589 15.12.9 Standard Error 590 15.13 Statistical Inference 593 15.14 Estimation 594 15.14.1 Estimator 594 15.15 Point Estimate and Interval Estimate 595 15.16 Confidence Interval for Population Mean and Proportion 596 15.16.1 Interval Estimation of Population Mean 596 15.16.2 Interval Estimation of Difference of Two Means 602 15.16.3 Interval Estimation for Population Proportion 603 List of Formulae 606 Self Practice Exercises 608

16. Test of Hypothesis 611–667 16.1 Introduction 611 16.2 Hypothesis and Hypothesis Test 611 16.3 Null Hypothesis and Alternative Hypothesis 612 16.4 Level of Significance and Critical Values 613 16.5 Critical and rejection region 614 16.6 Types of Tests — One and Two Tailed Test 614 16.7 Errors in Hypothesis Testing 615 16.8 Power of a Test and the Size Effect 616 16.9 Steps in Hypothesis Testing 617 16.10 Test of Significance 618 16.11 Hypothesis Testing for Large Samples (N > 30) 620 16.11.1 Test of Significance of Single Mean 620 16.11.2 p-Valve Approach to Test Hypothesis of Single Mean 624 16.11.3 Test of Significance for Difference between Two Means 627 16.11.4 Test of Significance of Single Proportion 630 16.11.5 Test of Significance for Difference between Proportions of Two Populations 635

Contents xxv

16.12 Hypothesis Testing for Small Samples (N < 30) 642 16.13 Student’s t-Distribution 642 16.13.1 Assumptions for Student’s t-Test 643 16.13.2 Properties of t-Distribution 643 16.13.3 Degrees of Freedom (d.f.) 643 16.13.4 Applications of t-Distribution 644 16.13.5 Test of Significance for Mean 645 16.13.6 Test of Significance for Difference between Two Means 650 16.13.7 Related Samples–Paired t-Test 654 16.14 F-Distribution 658 16.14.1 Assumptions for F-test 659 16.14.2 Properties of F-Distribution 659 List of Formulae 662 Self Practice Exercises 665

17. CHI-Square Test 668–685 17.1 Introduction 668 17.2 Chi-square Distribution 668 17.3 Properties of Chi-Square Distribution 669 17.4 Conditions for Applying 2 Test 669 17.5 The Chi-square test-statistic 669 17.5.1 Grouping When Frequencies Are Small 671 17.5.2 Applications of 2-Test 672 17.6 Test of Goodness-of-fit 672 17.7 Test of Independence : Contingency Analysis 676 17.7.1 2 × 2 Contingency Table 678 17.8 Yate’s Correction 678 17.9 Test for Population Variance 681 List of Formulae 683 Self Practice Exercises 683

18. Analysis of Variance (Anova) 686–706 18.1 Introduction 686 18.2 Computation of Analysis of Variance 687 18.3 One Way Classification 687 18.4 Test Procedure for One-way Analysis 689 18.5 Two Way Classification 694 18.6 Test Procedure for Two-way Analysis 695 Self Practice Exercises 703

19. Association of Attributes 707–740 19.1 Introduction 707

xxvi The Practice of Business Statistics

19.2 Classification of Data 707 19.3 Notations 708 19.4 Class and Class Frequencies 708 19.5 Order of Classes and Class Frequencies 708 19.6 Total Number of Class-frequencies 709 19.6.1 Ultimate Class Frequencies 710 19.7 Relationship Between Classes of Various Order 710 19.7.1 Two Attribute A and B 710 19.7.2 Two by Tow Table or Nine Square Table 710 19.8 Consistency of Data 712 19.9 Independence of Attributes 718 19.10 Association 720 19.10.1 Type of Association 721 19.11 Methods of measuring association 723 19.11.1 Frequency Method 723 19.11.2 Yule’s Coefficient of Association 725 19.11.3 Yule’s Coefficient of Colligation 730 19.11.4 Pearson’s Coefficient of Contingency Method 731 Self Practice Exercises 738

20. Statistical Decision Theory (SDT) 741–760 20.1 Introduction 741 20.2 Meaning and Scope 742 20.3 Decision Framework 743 20.4 Types of Decision Making 746 20.5 Decision Making Under Certainty 746 20.6 Decision Making Under Uncertainty (Without Probability) 747 20.7 Decision Making Under Risk (With Probability) 752 20.8 Decision Tree Analysis 754 20.8.1 Advantages of Decision Tree 755 Self Practice Exercises 757

21. Statistical Quality Control (SQC) 761–794 21.1 Introduction 761 21.2 Causes for Variation 761 21.3 Role of Sqc 762 21.4 Process and Product Control 762 21.5 Control Charts 762 21.6 Control Limits 763 21.7 Types of Control Charts 763 21.8 Control Charts for Variables 764

Contents xxvii

21.8.1 Control Chart For Mean or – Chart 764 21.8.2 R–Chart (Range Chart) 765 21.8.3 S–Chart 768 21.9 Control Chart for Attributes 769 21.9.1 C–Chart: A Chart for Average Number of Defects per Unit 770 21.9.2 p–Chart: A Chart for Proportion of Defectives 771 21.9.3 np–Chart: A Chart for Total Number of Defectives 775 21.10 Advantages of Statistical Quality Control 776 21.11 Limitations of Statistical Quality Control 777 21.12 Acceptance Sampling 777 21.13 Advantages of acceptance sampling 778 21.14 Types of Acceptance Sampling Plans 779 21.14.1 Single Sampling Plan 779 21.14.2 Double Sampling Plan 780 21.14.3 Multiple or Sequential Sampling Plan 781 21.15 The Operating Characteristic (OC) Curve 781 Miscellaneous Solved Examples 785 Self Practice Exercises 792

Appendix 795–804 T1. Areas under the Standard Normal Distribution 797 T2. Critical Value of 2 798 T3. Critical Value of t 799 T4. 1 % Points of F-Distribution 800 T5. 5 % Points of F-Distribution 801 T6. Binomial Coefficient 802 T7. Values of e–m 803 T8. Central Chart Factors 804

Index 805–809