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Quantitative Methods

© Emile Woolf International ii The Institute of Chartered Accountants of Pakistan

First edition published by Emile Woolf International Bracknell Enterprise & Innovation Hub Ocean House, 12th Floor, The Ring Bracknell, Berkshire, RG12 1AX United Kingdom Email: [email protected] www.emilewoolf.com

© Emile Woolf International, September 2013

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Quantitative Methods

© Emile Woolf International iv The Institute of Chartered Accountants of Pakistan

Page

17 Sampling and sampling distributions 411

18 Hypothesis testing 437

19 Chi-square testing 455

Appendix 465

Index 475

© Emile Woolf International v The Institute of Chartered Accountants of Pakistan

Assessment of Fundamental Competencies Quantitative Methods

S

Syllabus objective

and learning outcomes

ASSESSMENT OF FUNDAMENTAL COMPETENCIES

QUANTITATIVE METHODS

Objective

To develop the ability to apply quantitative methods and statistics to business problems.

Learning Outcome

On completion of this paper the candidate will be able to: 1 Understand basic mathematics to build a base for financial analysis and transform business problems into mathematical equations; 2 Apply financial mathematics to solve business problems; 3 Analyse business solutions and identify feasible, alternative, optimum and unbounded solutions using graphical methods;

4 Evaluate maximised profit, minimised cost and feasible manufacturing quantity by using calculus; 5 Analyse production planning cases and formulate solutions using matrices; 6 Present collected data using diagrams, charts and graphs and evaluate common measures of dispersion and central tendencies; 7 Evaluate the impact of inflation and rebase numbers using indices; 8 Use statistical methods in analyzing historical data for decision making and estimating future outcomes; 9 Explain and apply probability theory; 10 Explain sampling and explain and carry out tests of significance.

Syllabus objective and learning outcomes

© Emile Woolf International vii The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Mathematics Basic mathematics Exponential and logarithmic functions

2 LO1.1.1: Demonstrate adequate knowledge of laws of logarithm. LO1.1.2: Make use of logarithms in solving business problems. LO1.1.3: Perform calculations involving exponential and logarithmic functions. LO1.1.4: Analyse the behaviour of business problems involving exponential and logarithmic functions. Equation of straight line 2 LO1.2.1: Demonstrate adequate understanding of various forms of the equation of a straight line. Application of straight line in business and economics

2 LO1.3.1: Identify business situations where the equation of a straight line could be used. LO1.3.2: Use the equation of a straight line in relevant business problems. Simultaneous equation- linear and quadratic

2 LO1.4.1: Demonstrate adequate command of solving simple equations, including two simultaneous equations and quadratic equations. Coordinate system 2 LO1. 5 .1: Demonstrate understanding of the coordinate system and be able to prepare graphs of linear equations. System of linear inequalities and their graphical presentation

2 LO1. 6 .1: Demonstrate an understanding of linear inequalities. LO1. 6 .2: Demonstrate the graphical presentation of linear inequalities. Factorisation of equations including factorisation by completion of squares

2 LO1. 7 .1: Perform multiplication and division operations on linear and quadratic equations. LO1. 7 .2: Solve quadratic equations by factoring and by completing the square method.

Quantitative Methods

© Emile Woolf International viii The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Mathematics Basic mathematics (continued) Arithmetic progression 2 LO1. 8 .1: Identify situations where data is in arithmetic progression. LO1. 8 .2: Use arithmetic progression in business problems to calculate monthly instalments, first instalment, total amount paid and total time required for settlement of a loan etc. Geometric progression 2 LO1. 9 .1: Use geometric progression in relevant situations.

Financial mathematics Simple interest 2 LO 2. 1 .1: Calculate interest value using simple interest. Compound interest 2 LO 2. 2 .1: Calculate interest value using compound interest.

Present value 2 LO^2.^3 .1:^ Calculate the present^ value of a future cash sum using both a formula and tables. LO 2. 3. 2 : Calculate the net present value (NPV) of a project. LO 2. 3. 3 : Use NPVs to choose between mutually exclusive projects. Future values 2 LO2.4.1: Calculate future values using both simple and compound interest. Annuities 2 LO 2. 4 .1: Calculate the present value of an annuity using both a formula and tables. Internal rate of return 2 LO 2. 4 .1: Explain with examples the use of the internal rate of return of a project. Interpolation and perpetuities 2 LO 2. 4 .1: Calculate the present value of a perpetuity.

Quantitative Methods

© Emile Woolf International x The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Mathematics Matrices and determinants Fundamentals of matrices, addition, subtraction, multiplication, inverse of matrices

2 LO 5 .1.1: Demonstrate an adequate knowledge of matrix algebra (addition, subtraction and multiplication). LO 5 .1. 2 : Calculate the determinant, adjoint and inverse of a matrix. LO 5 .1. 3 : Make use of the properties of determinants while calculating determinants. Solution of simultaneous linear equations using Cramer’s Rule and Matrix Inverse Method

2 LO 5. 2 .1: Represent simultaneous linear equations in matrix form. LO 5. 2 .1: Solve simultaneous linear equations using Cramer’s Rule and Matrix Inverse Method.

Syllabus objective and learning outcomes

© Emile Woolf International xi The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Statistics Presentation and use of data Collection and tabulation of data 2 LO 6 .1.1: Classify different types of data. LO 6 .1. 2 : Perform data collection through various methods. LO 6 .1. 3 : Organise and summarise data and present it as a frequency distribution. Presentation through graphs, charts and diagrams, including stem and leaf display, box and whisker plot

2 LO 6. 2 .1: Present data using a simple bar chart, a multiple bar chart and a component bar chart. LO 6. 2. 2 : Construct pie charts, histograms, frequency polygons, ogives, stem and leaf displays and box and whisker plots. LO 6. 2. 3 : Analyse graphical representations of data. Measures of central tendencies and measures of dispersions

2 LO^6.^3 .1:^ Calculate various measures of central tendency such as mode, median, arithmetic, geometric and harmonic means. LO 6. 3. 2 : Analyse the advantages and disadvantages of various central tendency measures. LO 6. 3. 3 : Identify the characteristics and measures of dispersion. LO 6. 3. 4 : Use measures of dispersion, such as standard deviation or variance, to ascertain the degree of variation or variability in a distribution. Index numbers Index numbers, weighted index numbers, concept of purchasing power and deflation of income

2 LO 7 .1.1: Define the index number and its types. LO 7 .1. 2 : Use different formulae/methods to calculate various types of index number. LO 7 .1. 3 : Analyse the uses and limitations of index numbers. For example, use index numbers to deflate or inflate a series and explain the result.

Syllabus objective and learning outcomes

© Emile Woolf International xiii The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Probability and probability distribution (continued) Addition law for mutually exclusive and not mutually exclusive events

2 LO 9. 3 .1: Use the addition rule while calculating probabilities. LO9.3.2: Identify the difference between mutually exclusive and non-mutually exclusive events. Multiplicative laws for dependent and independent events

2 LO 9. 4 .1: Identify the difference between dependent and independent events. LO 9. 4. 2 : Use multiplication rule while calculating conditional probabilities. Binomial distribution 2 LO 9. 5 .1: Account for the assumptions that underlie the Binomial distribution. LO 9. 5. 2 : Demonstrate the use of Binomial distribution to calculate probabilities.

Poisson distribution 2 LO^9.^6 .1:^ Account for the properties of the Poisson distribution. LO 9. 6. 2 : Demonstrate the use of the Poisson distribution to calculate probabilities. Hyper-geometric distribution 2 LO 9. 7 .1: Identify situations where Hyper- geometric distribution could be used. LO9.7.2: Demonstrate the use of Hyper- geometric distribution to calculate probabilities. Normal distribution 2 LO 9. 8 .1: Demonstrate the use of normal distribution including the use of tables. LO9.8.2: Demonstrate the application of the normal distribution to calculate probabilities. Sampling and decision making Simple random sampling 2 LO10.1.1: Understand the terms, population and sample. LO10.1. 2 : Explain methods for selecting a simple random sample.

Sampling distribution of mean 2 LO10.^2 .1:^ Define and construct a sampling distribution of the sample means. LO10. 2. 2 : Calculate the mean and standard deviation of a sampling distribution of sampling means.

Quantitative Methods

© Emile Woolf International xiv The Institute of Chartered Accountants of Pakistan

Contents Level Learning Outcome

Sampling and decision making (continued) Standard error of mean 2 LO10. 3 .1: Calculate standard error of mean.

Sampling with and without replacement

2 LO10.^4 .1:^ Select^ an appropriate sampling technique for calculating the probabilities of sample means. Testing of hypothesis for population means, difference between population means and population proportion and difference between two population proportions

2 LO10. 5 .1: Demonstrate the use of hypothesis testing, significance testing and testing a hypothesis. LO10. 5. 2 : Perform hypothesis test of population means based on small and large samples. LO10. 5. 3 : Perform hypothesis tests of the difference between two population means based on small and large samples. LO10. 5. 4 : Perform hypothesis tests of the difference between two population proportions. LO10. 5. 5 : Select appropriate distributions i.e., z or t for constructing a confidence interval for a population mean. Single population variance based on test of Chi-square

2 LO10. 6. 1 : Use the Chi-square distribution to perform tests of goodness of fit and independence. Confidence interval for estimating population means, proportions and variance, and differences between proportions means, proportions and variance

2 LO10.7.1: Construct the confidence interval for population means and difference of means. LO10.7.2: Construct the confidence interval for population means and difference of proportion and variance. Problems of determination of sample size for the study of population mean and proportion

2 LO10.8.1: Calculate a sample size for an interval estimate of a population mean. LO10.8.2: Calculate a sample size for an interval estimate of a population proportion.

Quantitative Methods

© Emile Woolf International 2 The Institute of Chartered Accountants of Pakistan

INTRODUCTION

Learning outcomes

The overall objective of the syllabus is to develop the ability to apply quantitative methods and statistics to business problems.

Basic mathematics

LO 1 Understand basic mathematics to build a base for financial analysis and transform business problems into mathematical equations;

This chapter does not address any specific learning outcome. It introduces and revises basic mathematical techniques which are applied in later chapters.

Chapter 1: Elementary mathematical operations

© Emile Woolf International 3 The Institute of Chartered Accountants of Pakistan

1 ELEMENTARY MATHEMATICAL OPERATIONS

Section overview

 Basic techniques  Brackets  Parts of a whole – fractions  Operations involving fractions  Other ways of describing parts of a whole  Common uses of percentages

1.1 Basic techniques

Introduction

Both algebra and arithmetic deal with numbers. Both subjects employ the same fundamental operations and terminology. Arithmetic uses definite numbers so that problems can be solved to give a definite numerical solution. Algebra uses general expressions which arrive at a general result. Of course that does not stop values being found for the general expression in order to arrive at a numerical solution.

Definition An algebraic expression (expression) is a combination of symbols which stand for numbers in a mathematical operation. A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient.

An expression might contain the symbols of operation + or . When this is the

case the expression is separated by these into components known as terms.

Illustration: Terms (types of polynomial) 2x is an expression made up of a single term – a monomial. 2x + 3y is an expression made up of two terms – a binomial. 2x + 3y + z is an expression made up of three terms – a trinomial.

Algebra enables us to understand relationships between things by expressing them in a general way.

Chapter 1: Elementary mathematical operations

© Emile Woolf International 5 The Institute of Chartered Accountants of Pakistan

x changes by 2 then y changes by 4 and so on. The slope is 2 / 1 or 4 / 2 = 2. This is the coefficient of x. The slope of any straight line is always the coefficient of the independent variable. The chapter on calculus explains how to find the gradient of non-linear equations.

Signs

A number might be either positive (e.g.100) or negative (100). The following table summarises the rules on operations involving positive and negative numbers.

Addition and subtraction Multiplication Division +x + x = 2x a × b = ab a ÷ b = a/b x x = 2x a × b = ab a ÷ b = a/b x x = 0 a × b = ab a ÷ b = a/b x +x = 0 a × b = ab a ÷ b = a/b

Multiplication (or division) of two like signs give a positive outcome and multiplication (or division) of two unlike signs give a negative outcome.

Quantitative Methods

© Emile Woolf International 6 The Institute of Chartered Accountants of Pakistan

1.2 Brackets

Brackets are very important in the construction of complex expressions and equations as they indicate the order of mathematical operations.

Illustration: 2 × x + y

It is unclear whether this means that 2 should be multiplied by x and then y added to the sum or x should be added to y and then the sum multiplied by 2. Substituting values illustrates the problem. Let x = 5 and y =6. 2 × 5 = 10 + 6 = 16 5 + 6 = 11 × 2 = 22 Brackets clarify the situation by enclosing the part of an expression that should be operated as a whole. (2 × x) + y = 2x + y = 10 + 6 = 16 2 × (x + y) = 2(x + y) = 2(5 + 6) = 22

Solving equations often involves the need to rearrange terms and expressions in the equation and this might involve removal of the brackets. This is described as expanding the brackets. Terms within brackets might be added to or subtracted from another term.  When the terms in brackets are preceded by a + sign there is no change to the signs in the brackets.  When the terms in brackets are preceded by a  sign the signs in the brackets change.

a + (b + c) = a + b + c a + (b  c) = a + b  c a  (b + c) = a  b  c a  (b  c) = a  b + c

When terms in brackets are multiplied by a term outside the bracket each term within the brackets must be multiplied by the term outside.  When the outside term is +ve there is no change to the signs in the brackets.

 When the outside term is ve the signs in the brackets change.

x(a + b) = xa + xb x(a  b) = xa  xb x(a + b) = xa  xb x(a  b) = xa + xb