Business Mathematics and Statistics

BUSINESS MATHEMATICS AND STATISTICS part A-part B syllabus

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(UNIVERSITY OF DELHI) REVISED SYLLABUS FOR Courses of Reading For B.Com. for Academic Year 2006-2007 and onwards -----------------------------------------------------------------------------------B.Com. Part II Paper No. VI BUSINESS MATHEMATICS AND STATISTICS Duration : 3 hrs. Max. Marks: 75 Lectures: 75 Objective: The objective of this course is to familiarize students with the applications of  mathematics and statistical techniques in business decisions process. Notes: 1. Use of simple calculator is allowed. 2. Proofs of theorems / formulae are not required. 3. Trignometrical functions are not to be covered. PART – A: Business Mathematics (Marks: 25) Unit I: Matrices and Determinants 1.1 Definition of a matrix. Types of matrices. Algebra of matrices. 3 Lectures 1.2 Calculation of values of determinants up to third order. Adjoint of a matrix. Elementary row operations. Finding inverse of a matrix through adjoint and elementary row operations. Applications of matrices for solution to simple business and economic problems. 4 Lectures Unit II: Calculus 2.1 Mathematical functions and their t ypes – linear, quadratic, polynomial, exponential and logarithmic. Concepts of limit, and continuity of a function. 3 Lectures 2.2 Concept of differentiation. Rules of differentiation – simple standard forms

(involving one variable). 3 Lectures 2.3 Applications of differentiation – elasticities of demand and supply. Maxima and minima of functions (involving second or third order derivatives) relating to cost and revenue. 3 Lectures 2.4 Integration and its applications to business and economic situations. 3 Lectures Unit III : Basic Mathematics of Finance: 3.1 Simple and compound interest. Rates of interest – nominal, effective and continuous – and their inter-relationships. Compounding and discounting of a sum using different types of rates. 6 Lectures PART – B : Business Statistics (Marks : 50) Unit I: Descriptive Statistics for univariate data 1.1 Introduction to statistics. Preparation of frequency distributions including graphic presentations. 5 Lectures 1.2 Measures of Central Tendency 10 Lectures (a) Mathematical averages : Arithmetic mean, Geometric mean and Harmonic mean : Properties and applications. (b) Positional Averages : Mode and median and other partition values - quartiles, deciles, and percentiles (including graphic determination). 1.3 Measures of Variation: absolute and relative. Range, quartile deviation, mean deviation, standard deviation, and variance. 5 Lectures Unit II: Correlation and Regression Analysis 2.1 Correlation : Meaning, Correlation using scatter diagram. Karl Pearson's co-efficient of correlation: calculation and properties. 5 Lectures 2.2 Regression Analysis : Linear regression defined. Regression equations and estimation 5 Lectures Unit III: Index Numbers 3.1 Meaning and uses of i ndex numbers. Construction of index numbers: fixed and chain base; univariate and composite. Aggregative and average of relatives – simple and weighted. Tests of adequacy of index numbers. Construction of consumer price indices. 10 Lectures Unit IV: Time Series Analysis 4.1 Components of time series, additive and multiplicative models. 2 Lectures 4.2 Trend analysis. Finding trend by moving average method, Fitting of linear trend line using principle of least squares. 8 Lectures Suggested Readings: Business Mathematics: 1. E.T. Dowling, Mathematics for Economics, Schaum’s Outlines Series, McGraw Hill Publishing Co. 2. Mizrahi and Sullivan, Ma thematics for Business and Social Sciences, John Wiley and Sons

3. V.K. Kapoor, Essentials of Mathematics for Business and Economics, Sultan Chand and Sons. 4. J.K. Thukral, Mathematics for Business Studies, Mayur Publications 5. S.K. Singh & J. K. Singh, Business Mathematics, , Brijwasi Book distributors and publishers. 6. Zameeruddin, Business Mathematics by Vikas Publishing House (P) Ltd. Statistics: 1. Richard Levin and David S. Rubin, Statistics for Management, Prentice Hall of India, New Delhi. 2. M.R. Spiegel, Theory and Problems of Statistics, Schaum’s Outlines Series, McGraw Hill Publishing Co. 3. S.C. Gupta, Fundamentals of Statistics, Himalaya Publishing House. 4. S.P. Gupta and Archana Gupta, Elementary Statistics, Sultan Chand and Sons, New Delhi. 5. J. S. Chandan, Business Statistics, Vikas Publishing House. 6. B. N. Gupta, Statistics, Sahitya Bhawan Publishers and Distributers (P) Ltd.

The Application of Matrices to Business and Economics. Problem:

Suppose that the economy of a certain region depends on three industries: service, electricity and oil production. Monitoring the op erations of these three industries over a period of one year, we were able to come up with the following observations: 1- To produce 1 unit worth of service, the service industry must consume 0.3 units of its own  production, 0.3 units of electricity and 0.3 units of oil to run its operations. 2-To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of 

service, 0.1 units of its own production, and 0.5 units of oil. 3-Finally, the oil production company requires 0.3 units of service, 0.6 units of  electricity and 0.2 units of its own production to produce 1 unit of oil. Find the production level of each of these industries in order to satisfy the external and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system. Solution: 

Consider

the

1.  p1= production 2.  p2=

production

following

level for level

for

the the

variables: service

power-generating

plant

3.  p3= production level for the oil production company Since the model is closed, the total consumption of each industry must equal its total  production. This gives the following linear system: 0.3p1 + 0.3p2 + 0.3p3 = p1 0.4p1 + 0.1p2 + 0.5p3 = p2 0.3p1 + 0.6p2 + 0.2p3 = p3 The input matrix is: A= 0.3 0.3 0.3 0.4 0.1 0.5 0.3 0.6 0.2

industry (electricity)

and the above system can be written as (A-I)P=0. Note that this homogeneous system has infinitely many  solutions (and consequently a nontrivial solution) since each column in the coefficient matrix sums to 1. The augmented matrix of this homogeneous system is : -0.7 0.3 0.3 0 0.4 -0.9 0.5 0 0.3 0.6 -0.8 0 which can be reduced to : 1 0 -0.82 0 0 1 -0.92 0 0000 To solve the system, we let p3 =t (a parameter), then the general solution is :  p1= 0.82t  p2=0.92t  p3=t The values of the variables in this system must be nonnegative in order for the model to make sense. In other words, t ≥ 0. Taking t=100 for example would give the solution :  p1= 82 units  p2= 92 units  p3= 100 units. I chose an example of real life problem related to economics that shows how matrices can be effective and helpful when it comes to solving complicated problems. Matrices let us arrange numbers and get solutions to our problem in a quick and easy way !