Business Statistics. Fourth Edition. J. K. Sharma VIKAS PUBLISHING HOUSE PVT LTD. Professor Amity Business School, Amity University, Noida

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2 Business Statistics Fourth Edition J. K. Sharma Professor Amity Business School, Amity University, Noida VIKAS PUBLISHING HOUSE PVT LTD

3 VIKAS PUBLISHING HOUSE PVT LTD E-28, Sector-8, Noida (UP) India Phone: Fax: Registered Office: 576, Masjid Road, Jangpura, New Delhi India Website: Ahmedabad : 305, Grand Monarch, 100 ft, Shyamal Road, Near Seema Hall, Ahmedabad Ph Bengaluru : First Floor, N.S. Bhawan, 4th Cross, 4th Main, Gandhi Nagar, Bengaluru Ph , Chennai : E-12, Nelson Chambers, 115, Nelson Manickam Road, Aminjikarai Chennai Ph , Hyderabad : Aashray Mansion, Flat-G (G.F.), /8, Street No. 20, Himayath Nagar, Hyderabad Ph Fax Kolkata : 82, Park Street, Kolkata Ph Mumbai : 67/68, 3rd Floor, Aditya Industrial Estate, Chincholi Bunder, Malad (West), Mumbai Ph , Patna : Flat No. 101, Sri Ram Tower, Besides Chiraiyatand Over Bridge, Kankarbagh Main Road, Kankarbagh, Patna , (Bihar) Business Statistics ISBN: Fourth Edition 2014 Vikas is the registered trademark of Vikas Publishing House Pvt Ltd Copyright Vikas Publishing House Pvt Ltd., 2014 All rights reserved. No part of this publication which is material protected by this copyright notice may be reproduced or transmitted or utilized or stored in any form or by any means now known or hereinafter invented, electronic, digital or mechanical, including photocopying, scanning, recording or by any information storage or retrieval system, without prior written permission from the publisher. Information contained in this book has been published by VIKAS Publishing House Pvt Ltd and has been obtained by its Authors from sources believed to be reliable and are correct to the best of their knowledge. However, the Publisher and its Authors shall in no event be liable for any errors, omissions or damages arising out of use of this information and specifically disclaim any implied warranties or merchantability or fitness for any particular use. Disputes if any are subject to Delhi Jurisdiction only. Printed in India.

4 Contents Preface to the Fourth Edition Preface to the First Edition xv xvii CHAPTER 1 STATISTICS: AN OVERVIEW Reasons for Learning Statistics Growth and Development of Statistics Statistical Thinking and Analysis Statistics Defined Types of Statistical Methods Importance and Scope of Statistics Statistics and State Statistics in Economics Statistics in Business Management Statistics in Physical Sciences Statistics in Social Sciences Statistics in Medical Sciences Statistics and Computers Limitations of Statistics Statistics Does Not Study Qualitative Phenomena Statistics Does Not Study Individuals Statistics Can Be Misused How to Lie with Statistics? 8 Conceptual Questions 1A Need for Data Types of Data Principles of Measurement Sources of Data Primary Data Sources Secondary Data Sources 19 Conceptual Questions 1B 21 Chapter Concepts Quiz 22 CHAPTER 2 DATA CLASSIFICATION, TABULATION AND PRESENTATION Introduction Classification of Data Requisites of Ideal Classification Basis of Classification Organizing Data Using Data Array Frequency Distribution Methods of Data Classification Bivariate Frequency Distribution Types of Frequency Distributions 35

5 iv CONTENTS Conceptual Questions 2A 37 Self-practice Problems 2A 37 Hints and Answers Tabulation of Data Meaning and Definition Objectives of Tabulation Parts of a Table Types of Tables General and Summary Tables 42 Conceptual Questions 2B 46 Self-practice Problems 2B 46 Hints and Answers Graphical Presentation of Data Functions of a Graph Advantages and Limitations of Diagrams (Graphs) General Rules for Drawing Diagrams Types of Diagrams One-dimensional Diagrams Two-dimensional Diagrams Three-dimensional Diagrams Pictograms or Ideographs Cartograms or Statistical Maps Exploratory Data Analysis Stem-and-Leaf Displays 66 Conceptual Questins 2C 68 Self-practice Problems 2C 68 Hints and Answers 70 Formulae Used 70 Chapter Concepts Quiz 71 Review Self-practice Problems 72 Case Studies 74 CHAPTER 3 MEASURES OF CENTRAL TENDENCY Introduction Objectives of Averaging Requisites of a Measure of Central Tendency Measures of Central Tendency Mathematical Averages Arithmetic Mean of Ungrouped Data Arithmetic Mean of Grouped Data Some Special Types of Problems and Their Solutions Weighted Arithmetic Mean Advantages and Disadvantages of Arithmetic Mean 92 Conceptual Questions 3A 93 Self-practice Problems 3A 93 Hints and Answers Geometric Mean Combined Geometric Mean Weighted Geometric Mean Advantages, Disadvantages and Applications of G.M. 98 Conceptual Questions 3B 99 Self-practice Problems 3B 99 Hints and Answers Harmonic Mean Advantages, Disadvantages and Applications of H.M Relationship Among A.M., G.M. and H.M. 102 Self-practice Problems 3C 102 Hints and Answers Averages of Position 102

6 CONTENTS v Median Advantages and Disadvantages of Median Partition Values Quartiles, Deciles and Percentiles Graphical Method for Calculating Partition Values 106 Conceptual Questions 3C 110 Self-practice Problems 3D 110 Hints and Answers Mode Graphical Method for Calculating Mode Value Advantages and Disadvantages of Mode Value Relationship Between Mean, Median and Mode Comparison Between Measures of Central Tendency 115 Conceptual Questions 3D 116 Self-practice Problems 3E 116 Hints and Answers 118 Formulae Used 118 Chapter Concepts Quiz 119 Review Self-practice Problems 120 Hints and Answers 122 Case Studies 124 CHAPTER 4 MEASURES OF DISPERSION Introduction Significance of Measuring Dispersion Requisites for a Good Measure of Variation Classification of Measures of Dispersion Distance Measures Range Interquartile Range or Deviation 130 Conceptual Questions 4A 133 Self-practice Problems 4A 133 Hints and Answers Average Deviation Measures Mean Absolute Deviation Variance and Standard Deviation Mathematical Properties of Standard Deviation Chebyshev s Theorem Coefficient of Variation 149 Conceptual Questions 4B 151 Self-practice Problems 4B 152 Hints and Answers 154 Formulae Used 155 Chapter Concepts Quiz 155 Review Self-practice Problems 157 Hints and Answers 159 Case Studies 162 CHAPTER 5 SKEWNESS, MOMENTS AND KURTOSIS Introduction Measures of Skewness Relative Measures of Skewness 164 Conceptual Questions 5A 170 Self-practice Problems 5A 171 Hints and Answers Moments Moments About Mean Moments About Arbitrary Point 174

7 vi CONTENTS Moments About Zero (Origin) Relationship Between Central Moments and Moments About an Assumed Mean Moments in Standard Units Sheppard s Corrections for Moments Kurtosis Measures of Kurtosis 178 Conceptual Questions 5B 181 Self-practice Problems 5B 181 Hints and Answers 182 Formulae Used 182 Review Self-practice Problems 183 Hints and Answers 184 CHAPTER 6 FUNDAMENTALS OF PROBABILITY Introduction Concepts of Probability Random Experiment Sample Space Types of Event Definition of Probability Classical Approach Relative Frequency Approach Subjective Approach Fundamental Rules of Probability Glossary of Probability Terms Counting Rules for Determining the Number of Outcomes Multistep Experiments Combinations Permutations 192 Conceptual Questions 6A 193 Self-practice Problems 6A 194 Hints and Answers Rules of Probability and Algebra of Events Rules of Addition Rules of Multiplication 199 Self-practice Problems 6B 208 Hints and Answers Bayes Theorem 212 Self-practice Problems 6C 214 Hints and Answers 214 Formulae Used 215 Chapter Concepts Quiz 216 Review Self-practice Problems 217 Hints and Answers 219 Case Studies 222 CHAPTER 7 PROBABILITY DISTRIBUTIONS Introduction Probability Distribution Functions Cumulative Probability Distribution Function Expected Value and Variance of a Random Variable Expected Value Variance and Standard Deviation Properties of Expected Value and Variance 227 Conceptual Questions 7A 231 Self-practice Problems 7A 231

8 CONTENTS vii Hints and Answers Discrete Probability Distributions Binomial Probability Distribution 232 Conceptual Questions 7B 238 Self-practice Problems 7B 238 Hints and Answers Poisson Probability Distribution Negative Binomial Probability Distribution Multinomial Probability Distribution Hyper-geometric Probability Distribution 247 Conceptual Questions 7C 248 Self-practice Problems 7C 248 Hints and Answers Continuous Probability Distributions Normal Probability Distribution Function Approximation of Binomial and Poisson Distributions to Normal Distribution Uniform (Rectangular) Distribution Exponential Probability Distribution Gamma (Erlang) Distribution Beta Distribution 261 Conceptual Questions 7D 261 Self-practice Problems 7D 262 Hints and Answers 263 Formulae Used 264 Chapter Concepts Quiz 264 Review Self-practice Problems 266 Hints and Answers 267 CHAPTER 8 SAMPLING AND SAMPLING DISTRIBUTIONS Introduction Reasons of Sample Survey Types of Bias During Sample Survey Sampling and Non-sampling Errors Measurement of Sampling Error Population Parameters and Sample Statistics Principles of Sampling Principle of Statistical Regularity Principle of Inertia of Large Numbers Sampling Methods Probabilistic Sampling Methods Non-random Sampling Methods Choice of Sampling Methods Sampling Distributions Standard Error of Statistic Difference Between Population, Sample and Sampling Distributions 276 Conceptual Questions 8A Sampling Distribution of Sample Mean Sampling Distribution of Mean When Population Has Non-normal Distribution Sampling Distribution of Mean When Population Has Normal Distribution Sampling Distribution of Difference Between Two Sample Means 286 Self-practice Problems 8A 288 Hints and Answers Sampling Distribution of Sample Proportion Sampling Distribution of the Difference of Two Proportions 290

9 viii CONTENTS Self-practice Problems 8B 292 Hints and Answers 292 Formulae Used 293 Chapter Concepts Quiz 293 Review Self-practice Problems 295 Hints and Answers 295 CHAPTER 9 ESTIMATION AND CONFIDENCE INTERVALS Introduction Point Estimation Properties of a Point Estimator Drawback of Point Estimates Confidence Interval Estimation Interval Estimation of Population Mean (S Known) Interval Estimation for Difference of Two Means Interval Estimation of Population mean (s Unknown) 303 Self-practice Problems 9A 305 Hints and Answers Interval Estimation for Population Proportion 306 Self-practice Problems 9B 307 Hints and Answers Estimating Sample Size Sample Size for Estimating Population Mean Sample Size for Estimating Population Proportion Sample Size Determination for Finite Population 311 Conceptual Questions 9A 312 Self-practice Problems 9C 313 Hints and Answers 313 Formulae Used 313 Chapter Concepts Quiz 314 Review Self-practice Problems 315 Hints and Answers 317 CHAPTER 10 HYPOTHESIS TESTING Introduction Hypothesis and Hypothesis Testing Formats of Hypothesis Rationale for Hypothesis Testing General Procedure for Hypothesis Testing Direction of the Hypothesis Test Errors in Hypothesis Testing Power of a Statistical Test 325 Conceptual Questions 10A Hypothesis Testing for Population Parameters with Large Samples Hypothesis Testing for Single Population Mean Relationship Between Interval Estimation and Hypothesis Testing p-value Approach to Test Hypothesis of Single Population Mean Hypothesis Testing for Difference Between Two Population Means 335 Self-practice Problems 10A 339 Hints and Answers Hypothesis Testing for Single Population Proportion Hypothesis Testing for Difference Between Two Population Proportions Hypothesis Testing for a Binomial Proportion 344 Self-practice Problems 10B 346

10 CONTENTS ix Hints and Answers Hypothesis Testing for Population Mean with Small Samples Properties of Student s t-distribution Hypothesis Testing for Single Population Mean Hypothesis Testing for Difference of Two Population Means (Independent Samples) Hypothesis Testing for Difference of Two Population Means (Dependent Samples) 359 Self-practice Problems 10C 363 Hints and Answers Hypothesis Testing Based on F-Distribution Properties of F-Distribution Comparing Two Population Variances 367 Self-practice Problems 10D 369 Hints and Answers 370 Formulae Used 371 Chapter Concepts Quiz 372 Review Self-practice Problems 373 Hints and Answers 376 CHAPTER 11 CHI-SQUARE AND OTHER NON-PARAMETRIC TESTS Introduction Advantages and Limitations of Non-parametric Methods Chi-Square Distribution Properties of c2 Distribution Conditions for the Applications of c2 Test Chi-square Test-statistic Grouping of Small Frequencies Applications of c2 Test Contingency Table Analysis: Chi-Square Test of Independence 384 Self-practice Problems 11A 390 Hints and Answers Chi-square Test for Goodness-of-fit Yate s Correction for Continuity 395 Self-practice Problems 11B 398 Hints and Answers c 2 Test for Population Variance Coefficient of Contingency Chi-square Test of Homogeneity 403 Conceptual Questions 11A Sign Test for Paired Data Runs Test for Randomness Mann Whitney U-Test Wilcoxon Matched Pairs Test Kruskal Wallis Test 416 Self-practice Problems 11C 418 Hints and Answers 420 Formulae Used 420 Chapter Concepts Quiz 421 Review Self-practice Problems 422 Hints and Answers 424 CHAPTER 12 ANALYSIS OF VARIANCE Introduction Analysis of Variance Approach Testing Equality of Population Means: One-way Classification 429

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