McGraw-Hill UK

Quantitative Methods for Business and Management

Frank Dewhurst

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Table of contents

Front Matterix
Prefaceix
The intended audienceix
A justification for the bookix
The pedagogical stylex
How to use this bookx
The structure of this bookxi
The conventions used in this bookxi
Student CD-ROMxii
Online Learning Centrexii
Acknowledgementsxiii
Publisher’s acknowledgementsxiii
CD contentsxv
Examplesxv
Solutionsxv
Tablesxv
Data setsxv
Appendicesxv
OLC contentsxv
Guided Tourxvi
Technology to enhance learning and teachingxviii
Online Learning Centre (OLC)xviii
Resources for students include:xviii
Password-protected Lecturer resources include:xviii
Student CD-ROMxviii
For lecturers: Primis Content Centrexix
Study Skillsxix
Computing Skillsxix
An introduction to using calculators and Excelxx
Calculatorsxx
Computersxx
Spreadsheets and Microsoft Excelxx
Using Microsoft Excelxx
Figure i The main features of an Excel workbook.xxi
Configuring button bars in Excelxxi
Obtaining help in Excelxxi
Opening, naming and saving Excel filesxxi
Figure ii Configuring the Standard and Formatting button bars.xxii
Renaming worksheetsxxii
Figure iii Renaming a worksheet.xxii
Entering numbers, text and formulae into a cellxxiii
Figure iv Typing in the contents of a cell.xxiii
Editing and changing the contents of cellsxxiii
Filling cellsxxiv
Borders and coloursxxiv
Printing worksheetsxxiv
Closing and leaving Excelxxiv
Inbuilt functions and toolsxxiv
Data Analysis ToolPak and Solver Add-Insxxiv
Figure v Checking for the installation of the ToolPak and Solver.xxv
Figure vi The Add-Ins dialogue box.xxv
A note on the use of spreadsheets on the enclosed CD-ROMxxv
Part A: The foundations of Quantitative Methods1
Chapter 1 An introduction to Quantitative Methods3
Introduction3
Chapter objectives3
1.1 What is Quantitative Methods?4
Scientific methodology4
So what is Quantitative Methods?4
Figure 1.1.1 An iconic model of a typical scientific methodology.5
Help, I never understood maths at school5
1.2 Number systems6
The decimal system6
Example 1.2.16
Table 1.2.1 Expanded number notation for hundreds, tens and units.6
Condensed and expanded notation6
Table 1.2.2 General expanded notation.7
Table 1.2.3 Expanded notation for 18,105.06.7
Exponential notation7
Table 1.2.4 Exponential notation for tens, hundreds and higher multiples of ten.8
Table 1.2.5 Exponential notation for multiples of ten.8
Scientific notation8
Example 1.2.29
Other number systems9
Binary9
Table 1.2.6 Exponential form of binary10
Example 1.2.310
Table 1.2.7 Converting 10011 binary to decimal.10
Octal and hexadecimal10
Exercises 1.210
1.3 Truncating, rounding and estimating11
Rounding11
Example 1.3.111
Truncating12
Example 1.3.212
Estimating12
Example 1.3.312
Exercises 1.313
1.4 Basic arithmetic13
The sign of a number13
Addition and subtraction14
Multiplication14
Division14
The commutative law15
Comparison operators15
Rules of arithmetic15
Division by zero16
Exercises 1.416
1.5 Powers, roots and logarithms16
Exponents and powers16
Positive powers and exponents17
Example 1.5.117
Negative powers and exponents17
Example 1.5.217
Roots17
Reciprocal powers and exponents18
Example 1.5.318
Rules of exponents18
Logarithms to base 1019
Table 1.5.1 Some logarithms to base 10.19
Logarithms to other bases19
Table 1.5.2 Some logarithms to base 2.20
The natural logarithmic base (Euler’s constant)20
Rules of logarithms20
Table 1.5.3 Some squares, cubes, roots and logarithms.21
Exercises 1.521
1.6 Operator precedence21
Operator precedence21
Example 1.6.122
Example 1.6.222
Table 1.6.1 Operator precedence groupings.22
Table 1.6.2 BOMDAS mnemonic.22
Example 1.6.323
Exercises 1.623
1.7 Fractions, ratios and percentages24
Fractions24
Ratios24
Example 1.7.124
Percentages24
Example 1.7.225
Using percentages25
Example 1.7.325
Applications of percentages25
Mark-ups25
Example 1.7.426
Common factors26
Example 1.7.526
Marking down27
Example 1.7.627
Elasticity27
Example 1.7.728
Exercises 1.728
1.8 Units of measurement28
Units of measurement29
Table 1.8.1 Measurement unit abbreviations.29
Example 1.8.129
Example 1.8.230
Exercises 1.830
1.9 Chapter review30
Chapter 2 From numbers to symbols31
Introduction31
Chapter objectives31
2.1 Algorithms, variables and algebra32
Algorithms32
Example 2.1.132
Algorithm A133
Variables33
Algebra33
Example 2.1.234
Solving and simplifying algebraic problems34
Example 2.1.334
Example 2.1.435
Exercises 2.137
2.2 Simultaneous and systems of equations37
Simultaneous equations37
Example 2.2.137
Solving simultaneous equations38
Algorithm A238
Example 2.2.239
An improved algorithm39
Algorithm A339
Example 2.2.340
Example 2.2.440
Solving systems of equations40
Exercises 2.241
2.3 Scalars, vectors, matrices and arrays42
Scalars42
Vectors42
Example 2.3.142
Table 2.3.1 Quarterly sales for the clothing department of a Manchester store last year in £s.42
Matrices43
Example 2.3.243
Table 2.3.2 Quarterly sales for each department of a Manchester store last year in £s.43
Arrays43
Example 2.3.344
Array dimension44
Exercises 2.344
2.4 Subscripted variables and sigma notation45
Single subscripted variables and vector notation46
Sigma notation47
Example 2.4.147
General subscripted notation47
Example 2.4.248
Exercises 2.448
2.5 Set notation, set properties and Venn diagrams49
Set notation49
Example 2.5.149
Special sets and notation50
Venn diagrams50
Example 2.5.250
Figure 2.5.1 A simple Venn diagram.50
Set operations50
Equality of sets51
Subsets51
Proper subsets51
Union of sets51
Figure 2.5.2 The union of two sets.51
Example 2.5.352
Intersection of sets52
Figure 2.5.3 The intersection of two sets.52
Example 2.5.452
Disjoint sets52
Figure 2.5.4 Two disjoint sets.52
Example 2.5.553
Complement of a set53
Figure 2.5.5 The complement of a set.53
Example 2.5.653
de Morgan’s properties53
Figure 2.5.6 de Morgan’s properties.54
Example 2.5.754
Counting elements in sets54
Example 2.5.855
Figure 2.5.7 Customers buying strawberries and cream.55
Exercises 2.555
2.6 Counting, factorials, permutations and combinations56
Tree diagrams56
Example 2.6.156
Figure 2.6.1 A tree diagram for Example 2.6.1.56
The multiplication principle57
Factorials57
Example 2.6.258
Permutations58
Example 2.6.358
Figure 2.6.2 A permutation for Example 2.6.3.58
The permutation notation and formula59
Combinations59
Example 2.6.459
The combination notation and formula60
Exercises 2.660
2.7 Pascal’s triangle and binomial coefficients60
Pascal’s triangle61
Figure 2.7.1 Part of Pascal’s triangle.61
Binomial coefficients61
Example 2.7.161
The binomial theorem62
Example 2.7.262
Binomial identities62
Example 2.7.362
Example 2.7.462
Example 2.7.563
Exercises 2.763
2.8 Chapter review63
Chapter 3 Matrix algebra65
Introduction65
Chapter objectives65
3.1 Matrix addition and subtraction66
Addition and subtraction of vectors, matrices and arrays66
Matrix addition66
Example 3.1.166
Matrix subtraction66
Example 3.1.267
Exercises 3.167
3.2 Matrix multiplication68
Matrix multiplication68
Multiplying vectors, matrices and arrays by a scalar68
Example 3.2.168
Multiplying vectors and matrices68
Example 3.2.269
Multiplying a vector by a matrix69
Example 3.2.369
Multiplying a matrix by a matrix70
Example 3.2.470
General matrix multiplication71
Raising matrices to a power71
Example 3.2.572
Exercises 3.272
3.3 Special matrices73
Null vectors, matrices and arrays73
Example 3.3.173
Unit vectors, matrices and arrays74
Example 3.3.274
The identity matrix74
Example 3.3.374
Exercises 3.375
3.4 Special matrix operations75
The transpose matrix75
Example 3.4.176
The inverse matrix76
Example 3.4.276
The determinant of a matrix77
The determinant of a (2 × 2) matrix77
Example 3.4.377
Co-factors77
Example 3.4.478
The general determinant of a matrix78
Example 3.4.578
Finding the inverse matrix using co-factors79
Example 3.4.679
The inverse of a (2 × 2) matrix80
Example 3.4.780
Exercises 3.480
3.5 Using matrices to solve systems of equations81
The matrix form of a system of equations81
Example 3.5.181
The general matrix form81
Using the inverse matrix to solve systems of equations82
Example 3.5.282
Exercises 3.583
3.6 Chapter review84
Chapter 4 Elementary statistics85
Introduction85
Chapter objectives85
4.1 The meaning and nature of statistics86
Descriptive statistics86
Planning and designing a statistical study86
Types of data87
Nominal data87
Ordinal data88
Quantitative data88
Discrete data88
Continuous data88
Range of values and variables89
Figure 4.1.1 Ranges of some typical variables.89
Describing and presenting data89
Exercises 4.189
4.2 Sampling and data collection90
Sampling from a population90
Types of sampling91
Judgemental (non-random) sampling91
Quota (non-random) sampling91
Example 4.2.191
Systematic (pseudo-random) sampling91
Example 4.2.292
Simple random sampling92
Sampling with and without replacement92
Stratified (random) sampling92
Example 4.2.392
Cluster (random) sampling93
Example 4.2.493
Methods of collecting data94
Observation94
Experiments or tests94
Surveys94
Survey errors95
Guidelines for surveys and questionnaires95
Exercises 4.295
4.3 Presenting and describing small datasets96
Tables96
Table 4.3.1 A comparison of shopping costs.97
Pictures97
Scattergrams98
Example 4.3.198
Table 4.3.2 Monthly sales of sandals.98
Figure 4.3.1 Scattergram of sales of sandals at various prices.99
Figure 4.3.2 A revised scattergram of sales at various prices.99
Graphs99
Example 4.3.299
Figure 4.3.3 Prices of sandals over time (months).100
Figure 4.3.4 Sales of sandals each month.100
Figure 4.3.5 Sales at different prices.101
Figure 4.3.6 Price and sales over time.101
Pie charts101
Example 4.3.3102
Table 4.3.3 Sales by department.102
Table 4.3.4 Percentage sales and angles for data in Table 4.3.3.102
Figure 4.3.7 A basic pie chart for the data in Example 4.3.3.102
Figure 4.3.8 An exploded pie chart for the data in Example 4.3.3.102
Bar charts103
Example 4.3.4103
Table 4.3.5 Sales figures for each department of a retail store last year in £s.103
Figure 4.3.9 A multiple bar chart for the sales data in Example 4.3.4.103
Figure 4.3.10 A stacked bar chart for the sales data in Example 4.3.4.104
Pictograms104
Figure 4.3.11 Millions of hectares under trees in each region of the UK.104
Figure 4.3.12 Millions of hectares under trees in each region of the UK.104
Exercises 4.3105
4.4 Presenting and describing large datasets106
Data reduction106
Example 4.4.1107
Table 4.4.1 Daily sales of baked beans in a London store.107
Table 4.4.2 Classes for Example 4.4.1.107
Tallies107
Table 4.4.3 Tally for Example 4.4.1.108
Frequency tables108
Table 4.4.4 Frequency distribution for Example 4.4.1.108
Cumulative frequency108
Table 4.4.5 Cumulative frequency distribution for Example 4.4.1.108
Percentage cumulative frequency109
Table 4.4.6 % frequency and % cumulative frequency distributions for Example 4.4.1.109
Histograms109
Figure 4.4.1 Histogram for Example 4.4.1.109
Example 4.4.2110
Table 4.4.7 Sales of baked beans at a Birmingham store.110
Figure 4.4.2 Histogram for Example 4.4.2.110
Figure 4.4.3 Alternative histogram for Example 4.4.2.111
Ogives111
Figure 4.4.4 Ogive for Example 4.4.1.111
Frequency polygons111
Figure 4.4.5 Frequency polygon for Example 4.4.1.112
Lorenz curves112
Example 4.4.3112
Table 4.4.8 Incomes and population last year.112
Table 4.4.9 Cumulative percentages for Example 4.4.3.112
Figure 4.4.6 Lorenz curve for Example 4.4.3.113
Exercises 4.4113
4.5 Chapter review114
Chapter 5 Summary statistics115
Introduction115
Chapter objectives115
5.1 Measures of location116
Measures of central location116
The arithmetic mean116
Example 5.1.1117
Table 5.1.1 The sum of employee ages.117
The median118
Example 5.1.2118
Table 5.1.2 The ages of the employees in Example 5.1.1 in rank order.118
The mode118
Example 5.1.3118
Measures of central location for grouped data118
The group arithmetic mean118
Example 5.1.4119
Table 5.1.3 Calculations for the group mean of Example 5.1.4.119
The group median119
Example 5.1.5120
Table 5.1.4 Cumulative frequencies for Example 4.4.2.120
The group mode120
Figure 5.1.1 Estimating the group mode.120
Example 5.1.6121
Table 5.1.5 Frequency distribution for Example 4.4.2.121
Figure 5.1.2 Histogram for Example 4.4.2.121
Which measure of central location?122
Figure 5.1.3 A symmetrical distribution.122
Figure 5.1.4 A positively skewed distribution.123
Figure 5.1.5 A negatively skewed distribution.123
The weighted mean123
Example 5.1.7124
Exercises 5.1124
Figure 5.1.6 Histogram of sales.124
5.2 Measures of non-central location125
Quartiles125
Example 5.2.1126
Table 5.2.1 The ranked data for Example 5.1.1.126
Example 5.2.2126
Table 5.2.2 Cumulative frequencies for Example 4.4.2.127
Percentiles127
Example 5.2.3127
Table 5.2.3 Cumulative frequencies for Example 4.4.2.127
Exercises 5.2128
5.3 Measures of variation129
Measures of variation129
Figure 5.3.1 Histogram with class limits from 5 to 50.129
Figure 5.3.2 Histogram with class limits from 10 to 40.129
Range130
Interquartile range130
Box and whisker plots130
Example 5.3.1130
Figure 5.3.3 Box and whisker plot.130
Deviations from the mean131
Example 5.3.2131
Table 5.3.1 Deviation from the mean.131
Mean Absolute Deviation (MAD)132
Example 5.3.3132
Table 5.3.2 Calculations for MAD for Example 4.4.2.132
Variance133
Standard deviation133
Sample variance and sample standard deviation133
Example 5.3.4134
Table 5.3.3 Group variance and standard deviation.135
Coefficient of variation135
Example 5.3.5135
Coefficient of quartile deviation136
Exercises 5.3136
5.4 Index numbers137
Index numbers137
Example 5.4.1137
Table 5.4.1 A simple price index.137
Weighted price indices138
The Retail Price Index138
Table 5.4.2 Part of an RPI.138
Example 5.4.2138
Laspeyre’s indices139
Example 5.4.3139
Table 5.4.3 Quantities purchased for goods A, B and C over 4 years.139
Table 5.4.4 Prices paid for goods A, B and C over 4 years.139
Table 5.4.5 Laspeyre’s quantity index for Example 5.4.3.140
Table 5.4.6 Laspeyre’s price index for Example 5.4.3.140
The Paasche indices140
Example 5.4.4141
Table 5.4.7 The Paasche quantity index for Example 5.4.4.141
Table 5.4.8 The Paasche quantity index for Example 5.4.4.141
Exercises 5.4142
5.5 Chapter review143
Part A Closing comments144
Case A1 Tops ‘n’ Bottoms145
Case A2 CompRus Ltd146
Case A3 Joowel’s Store147
Part B: Models and analysis for business and management149
Chapter 6 Probability and statistical models151
Introduction151
Chapter objectives151
6.1 An introduction to probability152
The sample space and events152
Example 6.1.1152
Figure 6.1.1 Service-level outcomes for Example 6.1.1.152
Set notation for events152
Example 6.1.2152
Figure 6.1.2 Good and poor service outcomes.153
Measuring probability153
Probability as relative frequency154
Probability notation154
Example 6.1.3154
Table 6.1.1 Frequency distribution for Example 6.1.3.154
Table 6.1.2 Probabilities of monthly sales for Example 6.1.3.154
Long-run relative frequency155
The properties of probability155
Types of event155
Independent events155
Mutually exclusive events156
Figure 6.1.3 Two mutually exclusive events.156
Non-mutually exclusive independent events156
Figure 6.1.4 Two non-mutually exclusive events.157
Conditional probabilities157
Dependent events157
Bayes’ theorem158
Example 6.1.4158
Probability distributions159
Figure 6.1.5 A probability distribution for Example 6.1.3.159
The mean, variance and standard deviation of a distribution159
Example 6.1.5160
Table 6.1.3 Calculations for mean and variance for Example 6.1.3.160
Discrete and continuous probability distributions161
Figure 6.1.6 A continuous probability distribution.161
Random variables161
Example 6.1.6161
Mean, variance and standard deviation of discrete random variables162
The area under a probability distribution162
Cumulative probability distributions162
Figure 6.1.7 The area under a probability distribution for Example 6.1.3.163
Table 6.1.4 Cumulative probabilities for Example 6.1.3.163
Figure 6.1.8 A cumulative probability distribution for Example 6.1.3.163
Example 6.1.7163
Theoretical probability distributions164
Exercises 6.1164
6.2 The Binomial distribution165
The probabilities of outcomes165
Example 6.2.1165
The Binomial distribution165
Figure 6.2.1 Some Binomial distributions.166
Calculating Binomial probabilities166
Example 6.2.2166
Figure 6.2.2 The Binomial distribution for Example 6.2.2.167
Calculating probabilities of several events occurring167
Example 6.2.3167
Figure 6.2.3 P(X ≤ 2) and P(X > 2) for Example 6.2.3.168
Figure 6.2.4 P(X < 2) and P(X > 2) for Example 6.2.3.169
Figure 6.2.5 P (1 ≤ X ≤ 3) for Example 6.2.3.169
Cumulative Binomial probabilities169
Example 6.2.4169
Table 6.2.1 Probabilities and cumulative probabilities for Example 6.2.4.170
Figure 6.2.6 The cumulative Binomial distribution for Example 6.2.4.170
Calculating probabilities from cumulative tables170
Example 6.2.6170
The mean, variance and standard deviation of the Binomial distribution171
Example 6.2.7171
Table 6.2.2 Calculations for mean and variance for Example 6.2.2.171
Using the Binomial distribution as a model171
Example 6.2.8172
Table 6.2.3 Empirical and Binomial probabilities for Example 6.1.3.172
Figure 6.2.7 Empirical and Binomial distributions for Example 6.1.3.172
Large numbers of trials and small probabilities173
Example 6.2.9173
Exercises 6.2173
6.3 The Poisson distribution174
Calculating Poisson probabilities174
Figure 6.3.1 Some typical Poisson distributions.175
The mean, variance and standard deviation175
Applications of the Poisson distribution175
Example 6.3.1176
Example 6.3.2176
Table 6.3.1 Poisson probabilities for Example 6.3.2.176
Figure 6.3.2 Poisson probabilities for Example 6.3.2.177
Cumulative Poisson probabilities177
Example 6.3.3177
Using a Poisson distribution as a model177
Example 6.3.4178
Table 6.3.2 Empirical and Poisson probabilities for Example 6.3.4.178
Figure 6.3.3 Empirical and Poisson distribution for Example 6.3.3.178
Exercises 6.3178
6.4 The Exponential distribution179
Exponential and other continuous probability distributions179
Figure 6.4.1 Some Exponential distributions.180
The mean and standard deviation of the Exponential distribution180
Example 6.4.1180
Example 6.4.2181
Exercises 6.4181
6.5 The Normal distribution182
Calculating Normal probabilities182
Figure 6.5.1 Some Normal distributions.182
The Standard Normal distribution183
Standard Normal tables183
Example 6.5.1183
Figure 6.5.2 Distribution of weights of cans.183
Figure 6.5.3 The empirical and the Standard Normal distributions.184
Table 6.5.1 Part of the cumulative Standard Normal distribution table.185
Figure 6.5.4 Area under the left-hand tail.185
Example 6.5.2185
Figure 6.5.5 Area under the right-hand tail of the distribution.186
Example 6.5.3186
A note on the use of Standard Normal tables186
Figure 6.5.6 Area between two values.187
Figure 6.5.7 Using one-tail Standard Normal tables.187
Approximating Binomial probabilities using the Normal distribution187
Exercises 6.5188
6.6 Chapter review188
Chapter 7 An introduction to inferential statistics189
Introduction189
Chapter objectives189
7.1 Estimating and confidence intervals190
Estimating the population mean190
Example 7.1.1190
Table 7.1.1 Two samples of the weights (grams) of 25 cans.190
The sampling distribution of sample means191
Example 7.1.2191
Table 7.1.2 The mean weights (grams) of 50 samples of baked bean cans.191
Table 7.1.3 Frequency distribution of means.191
Figure 7.1.1 A sampling distribution of means for Example 7.1.2.192
The sampling distribution of means for large numbers of samples192
Figure 7.1.2 Distribution of the weights (grams) of 5,000 cans of baked beans.192
The mean of the sampling distribution of means192
Example 7.1.3192
The standard deviation of the sampling distribution of the means193
Example 7.1.4193
The Central Limit Theorem193
Modelling the sampling distribution of the means by the Normal distribution193
Figure 7.1.3 The population and sampling distribution of the means.194
Confidence intervals194
Figure 7.1.4 The probability that zx¯ lies in the interval −1.96 to 1.96.195
Table 7.1.4 Part of the Cumulative Standard normal table showing the value of z for 0.025.195
Example 7.1.5196
Calculating confidence intervals for the population mean196
Example 7.1.6196
Figure 7.1.5 The probability that z lies in the interval −1.645 to 1.645.197
One-tailed confidence intervals for the population mean197
Figure 7.1.6 Comparing one- and two-tailed 95% confidence intervals.198
Example 7.1.7198
Confidence intervals for small samples (the t-distribution)199
Figure 7.1.7 Student’s t-distribution for 5, 15 and 30 degrees of freedom.199
Example 7.1.8199
Estimating other parameters of a population200
Exercises 7.1200
7.2 An introduction to hypothesis testing200
Hypothesis testing200
Example 7.2.1201
Levels of significance201
Figure 7.2.1 Acceptance and rejection regions for 5% level of significance.201
Example 7.2.2202
Errors in hypothesis testing202
Figure 7.2.2 Type I and Type II errors.203
One-tailed hypothesis tests203
Figure 7.2.3 One-tailed hypothesis tests for 5% level of significance.203
Example 7.2.3204
Testing for the difference in two means204
Example 7.2.4204
Hypothesis testing using small samples205
Testing other hypotheses205
Exercises 7.2205
7.3 Non-parametric tests206
The Chi-squared distribution206
Figure 7.3.1 Chi-squared distributions.206
The Chi-squared hypothesis test206
Table 7.3.1 Calculating the value of x2.207
Example 7.3.1207
Table 7.3.2 Annual industrial fatalities.207
Table 7.3.3 Calculating the value of X2.208
Example 7.3.2208
Table 7.3.4 Frequency and probability distributions for Example 6.2.8.208
Table 7.3.5 Calculating the value of Chi-squared for Example 7.3.2.208
Hypothesis tests of association209
Example 7.3.3209
Table 7.3.6 Body Mass Index.209
Table 7.3.7 Row and column totals for Table 7.3.6.209
Table 7.3.8 Expected frequencies for Table 7.3.6.210
Table 7.3.9 Calculating X2 for Example 7.3.3.210
Exercises 7.3210
7.4 Chapter review211
Chapter 8 Modelling relationships212
Introduction212
Chapter objectives212
8.1 Relationships, functions and equations213
Example 8.1.1213
Table 8.1.1 Retail sales over a 6-month period for Example 8.1.1.213
Figure 8.1.1 A scattergram of sales against unit price for Example 8.1.1.213
Functions and relationships214
Figure 8.1.2 Scattergram for Example 8.1.1 showing a line passing through all points.214
Exercises 8.1215
8.2 Linear functions216
The linear function216
Figure 8.2.1 A graph of the linear function y = 2 x + 5.216
Figure 8.2.2 A graph of the linear function y = −2x + 5.217
Finding the equation of a line217
Example 8.2.1217
Finding the equation of a line from a graph217
Example 8.2.2218
Figure 8.2.3 The slope and intercept of a line joining the points.218
Finding the equation of a line from the data218
Example 8.2.3218
Exercises 8.2219
8.3 Least-squares linear regression220
A best-fit line220
Example 8.3.1220
Table 8.3.1 Manufacturing costs and output over 5 years.220
Figure 8.3.1 A possible best-fit line for Example 8.3.1.221
Quantifying the best-fit line221
MAE and MSE221
Example 8.3.2222
Table 8.3.2 The calculation of MAE and MSE for Example 8.3.1.222
Least-squares linear regression222
Example 8.3.3223
Table 8.3.3 Calculations for the least-squares regression line for Example 8.3.3.223
Exercises 8.3223
8.4 Appropriateness and correlation224
How good is the model?224
The coefficient of determination225
The correlation coefficient225
Example 8.4.1225
Table 8.4.1 Calculations for the coefficient of determination for Example 8.4.1.225
Appropriateness226
Example 8.4.2226
Figure 8.4.1 A typical non-linear demand function.227
Exercises 8.4227
8.5 Chapter review228
Chapter 9 Non-linear and multivariate relationships229
Introduction229
Chapter objectives229
9.1 Quadratic functions230
The quadratic function230
Figure 9.1.1 Some typical parabolas (the line of symmetry is shown dashed).231
Example 9.1.1231
Table 9.1.1 Sales and revenue at various prices for Example 9.1.1.231
Figure 9.1.2 A revenue curve for Example 9.1.1, showing its roots and turning point.232
Fitting quadratic functions232
Example 9.1.2232
Table 9.1.2 Profit and production output over 3 years for Example 9.1.2.233
Figure 9.1.3 A scattergram of profit against output for Table 9.1.2.233
Figure 9.1.4 An exact-fitting quadratic profit function for Example 9.1.2.234
Best-fit quadratic functions234
Example 9.1.3234
Table 9.1.3 Monthly output and profits for Example 9.1.3.235
Figure 9.1.5 Best-fit quadratic model of profit for Example 9.1.3.235
Exercises 9.1235
9.2 Polynomial functions236
Linear and quadratic functions236
Table 9.2.1 Linear and quadratic functions.236
Polynomial functions237
Table 9.2.2 Part of the polynomial family of functions.237
Figure 9.2.1 The graph and features of a polynomial function.238
Exercises 9.2238
9.3 Hyperbolic functions238
The basic rectangular hyperbola239
Figure 9.3.1 The graph of a simple rectangular hyperbola.239
Rectangular hyperbolae239
Figure 9.3.2 Graph of equation (9.3.2).240
Figure 9.3.3 Graph of equation (9.3.3).240
A general rectangular hyperbola240
Figure 9.3.4 A general rectangular hyperbola.241
Applications of rectangular hyperbolae241
Example 9.3.1241
Figure 9.3.5 Average costs per unit for Example 9.3.1.242
Example 9.3.2242
Table 9.3.1 Sales over 3 months.242
Figure 9.3.6 Sales forecasts over 12 months for Example 9.3.2.244
Best-fit hyperbolic function244
Example 9.3.3244
Table 9.3.2 Student times for word-processing test.244
Figure 9.3.7 Scattergram of test times for Example 9.3.3.245
Table 9.3.3 Least-squares regression calculations for Example 9.3.3.245
Figure 9.3.8 Scattergram and hyperbolic function for Example 9.3.3.246
Exercises 9.3246
9.4 Exponential functions247
A simple exponential function247
Example 9.4.1248
Figure 9.4.1 The growth in the number of agents over time.248
Table 9.4.1 Number of agents per month.248
Features of the simple exponential function249
Figure 9.4.2 Features of a simple exponential function.249
A standard form for exponential functions249
Figure 9.4.3 Graph of a simple exponential function.250
The exponential family250
Applications of exponential functions250
Example 9.4.2250
Table 9.4.2 Sales over time.250
Figure 9.4.4 An exponential sales function for Example 9.4.2.252
Best-fit exponential functions252
Example 9.4.3252
Table 9.4.3 Least-squares regression calculations for Example 9.4.3.253
Figure 9.4.5 An exponential model for Example 9.4.3.253
Other exponential functions253
Example 9.4.4254
Figure 9.4.6 Graph of GNP over time.254
Exercises 9.4255
9.5 Multivariate models255
Multivariate functions256
Linear multivariate functions256
Figure 9.5.1 Part of a bivariate linear function.256
Example 9.5.1257
Table 9.5.1 Unit production costs for five products.257
Non-linear multivariate functions257
Example 9.5.2257
Figure 9.5.2 A sketch of R=−5p12−10p22+100p1+200p2.258
Multivariate quadratic functions258
Figure 9.5.3 Bivariate quadratic with minima in both variables.258
Figure 9.5.4 A bivariate quadratic having a saddle.259
Exercises 9.5259
9.6 Chapter review259
Chapter 10 Analysing models261
Introduction261
Chapter objectives261
10.1 Graphical and numerical analysis262
Figure 10.1.1 Typical features of a graph.263
Trend263
Predictions263
Example 10.1.1263
Figure 10.1.2 Graph of linear total cost function.263
Intercepts264
Roots264
Example 10.1.2264
Figure 10.1.3 Intercept and roots of a rectangular hyperbola.265
Example 10.1.3265
Figure 10.1.4 Intercept and roots of a quadratic.266
Example 10.1.4266
Figure 10.1.5 Roots of a cubic.266
Table 10.1.1 Finding a root in the interval 5 < x < 6.267
Finding turning points267
Finding asymptotes267
Example 10.1.5267
Figure 10.1.6 The asymptotes of equation (10.1.7).268
Finding optima268
Multivariate functions268
Example 10.1.6269
Figure 10.1.7 Graph of −5p12−10p22+100p1+200p2 holding p2 constant.269
Figure 10.1.8 Graph of −5p12−10p22+100p1+200p2 holding p1 constant.269
Figure 10.1.9 The contours of −5p12−10p22+100p1+200p2.269
Figure 10.1.10 Part of the graph of R=−5p12−10p22+100p1+200p2.270
Exercises 10.1270
10.2 Marginal analysis and differentiation271
Marginal analysis271
Example 10.2.1272
Table 10.2.1 Marginal profits.272
Figure 10.2.1 Profit and marginal profit.273
Derivatives273
Differentiation and differential calculus274
Figure 10.2.2 A function with a turning point between two integers.274
Differentiation from first principles274
Example 10.2.2274
Figure 10.2.3 Small changes.275
Rules of differentiation276
The Power Rule276
Example 10.2.3276
The Exponential Rule278
Example 10.2.4278
The Product Rule278
Example 10.2.5279
The Quotient Rule280
Example 10.2.6280
The Chain Rule282
Example 10.2.7282
Exercises 10.2283
10.3 Elasticity284
Arc Elasticity284
Example 10.3.1285
Figure 10.3.1 Arc Price Elasticity of demand.285
Point Elasticity286
Example 10.3.2286
Exercises 10.3287
10.4 Turning points288
Locating turning points288
Figure 10.4.1 The slope is zero at a turning point.288
Example 10.4.1289
The type of turning point289
Figure 10.4.2 The slope around a maximum.290
Figure 10.4.3 The slope around a minimum.290
Example 10.4.2291
Exercises 10.4291
10.5 Integration292
Indefinite integration292
Example 10.5.1293
The Power Rule of integration293
Example 10.5.1 (revisited)293
Definite integration294
Example 10.5.2294
Figure 10.5.1 The profit function of all gas suppliers.294
Figure 10.5.2 The approximate area under the curve.295
Table 10.5.1 Approximate area under a curve.295
Exercises 10.5296
10.6 Partial differentiation and applications297
Partial differentiation297
Example 10.6.1298
Approximating the total derivative299
Example 10.6.2300
Partial Point elasticity300
Example 10.6.3301
Stationary points301
Example 10.6.4302
Exercises 10.6303
10.7 Chapter review304
Chapter 11 An introduction to time series305
Introduction305
Chapter objectives305
11.1 Classical time-series analysis306
Trend306
Figure 11.1.1 Time series with constant trend.306
Figure 11.1.2 Linear upward-sloping (increasing) trend.306
Figure 11.1.3 Non-linear asymptotic trend.307
Figure 11.1.4 Oscillating and little or no apparent trend.307
Seasonal variation307
Figure 11.1.5 Seasonal variation.308
Figure 11.1.6 Seasonal variation.308
Cyclical variation308
Irregular variation308
Visual inspection308
Example 11.1.1309
Table 11.1.1 Monthly tyre sales over 5 years.309
Figure 11.1.7 Graph for Example 11.1.1.309
Exercises 11.1309
11.2 Classical time-series models310
Time-series models310
Calculating values of the components311
Moving averages311
Figure 11.2.1 Constructing a 3-period moving average.312
Table 11.2.1 A 3-period moving average.312
Example 11.2.2312
Table 11.2.2 A 12-period moving average for Example 11.1.1.313
Figure 11.2.2 The smoothing effect of a moving average.313
Calculating seasonal components314
Example 11.2.3314
Table 11.2.3 Ratio of actual data to moving averages.314
Table 11.2.4 Ratio to MA seasonal indices for Example 11.2.3315
Table 11.2.5 Seasonal indices for Example 11.2.3.315
Calculating the trend, cyclical and irregular components316
Example 11.2.4316
Table 11.2.6 Deseasonalised data for Example 11.2.3.316
Figure 11.2.3 Graph of deseasonalised data.317
Modelling the trend317
Example 11.2.5317
Figure 11.2.4 The trend in the deseasonalised data.317
Exercises 11.2318
11.3 Classical time-series forecasting318
Forecasting319
Testing and validating forecasts319
Example 11.3.1319
Table 11.3.1 Seasonal indices for Example 13.3.1.320
Table 11.3.2 Forecasts for year 4.320
Figure 11.3.1 Year 4 forecasts and actuals.320
Table 11.3.3 Forecast error.321
Table 11.3.4 Chi-squared test.321
Exercises 11.3322
11.4 Exponential smoothing322
The Simple Exponential Smoothing (SES) model322
Example 11.4.1323
Table 11.4.1 Sales data for Example 11.4.1.324
Table 11.4.2 Exponential smoothing for Example 11.4.1.325
Figure 11.4.1 Actual sales and exponential smoothing for Example 11.4.1.325
Choosing the smoothing factor326
The Holt–Winters exponential smoothing model326
Example 11.4.2327
Table 11.4.3 Initial estimates.327
Table 11.4.4 Forecasts (rounded to the nearest integer) for years 4 and 5.328
Figure 11.4.2 Actual versus forecasts for years 4 and 5.329
Forecasting many periods ahead from the current period329
Example 11.4.3329
Exercises 11.4329
11.5 Chapter review330
Chapter 12 Models for finance and accounting331
Introduction331
Chapter objectives331
12.1 Simple interest and compound growth332
Interest and growth332
Simple interest332
Example 12.1.1332
The simple interest formula332
Figure 12.1.1 Value of money over time for simple interest.333
Example 12.1.2333
Compound interest and growth333
Example 12.1.3333
Figure 12.1.2 The value of money over time for Example 12.1.3.334
The basic compound growth formula334
Figure 12.1.3 Future values for basic compound growth.334
Example 12.1.4335
Exercises 12.1335
12.2 Multiple compounding, AER and continuous compounding335
Monthly compounding336
Example 12.2.1336
The monthly compounding formula336
Figure 12.2.1 Monthly compounding for 1 year.336
Figure 12.2.2 Monthly compounding over Y years.336
Multiple compounding337
Example 12.2.2337
The Annual Equivalent Rate337
Example 12.2.3337
Example 12.2.4339
The Annual Percentage Rate (APR)339
Continuous compounding339
Example 12.2.5339
Exercises 12.2340
12.3 Discounting and other related applications of compound growth340
Discounting340
Example 12.3.1341
Calculating the growth/discounting horizon and growth/discounting rates341
Example 12.3.2342
Reducing-balance depreciation342
Example 12.3.3342
Exercises 12.3342
12.4 Savings, endowments and sinking funds343
Saving schemes343
Example 12.4.1344
Figure 12.4.1 The future values for payments made in Example 12.4.1.344
The future value of saving schemes344
Figure 12.4.2 The future values for payments made at the end of each period into a saving scheme.344
Calculating the future value of a savings scheme as a geometric series345
Example 12.4.2345
Saving schemes with payments at the start of a period346
Example 12.4.3346
Figure 12.4.3 The future values for payments made in Example 12.4.3.346
The future value of savings with payments at the start of a period347
Figure 12.4.4 The future values for payments made at the start of a period into a savings scheme.347
Example 12.4.4347
Exercises 12.4348
12.5 Loans, mortgages and annuities348
Loans and mortgages349
Interest-only loans and mortgages349
Example 12.5.1349
Repayment loans and mortgages349
Example 12.5.2350
Figure 12.5.1 The end-of-year outstanding debt for Example 12.5.1.350
Formulae for repayment loans and mortgages350
Figure 12.5.2 The outstanding debt at the end of each period.350
Example 12.5.3352
Rip-off charges352
Example 12.5.4353
Endowment loans and mortgages353
Example 12.5.5353
Annuities354
Example 12.5.6354
Exercises 12.5355
12.6 Chapter review355
Part B Closing comments356
Case B1 NHT Beds357
Case B2 Techno Ltd357
Case B3 Mr and Mrs Lean358
Part C: Modelling and analysing decisions359
Chapter 13 An introduction to decision analysis361
Introduction361
Chapter objectives361
13.1 Decisions under certainty, uncertainty and risk362
Decision–making362
Problem maps362
Figure 13.1.1 Problem map for a student of business studies.362
Modelling decision problems363
Pay–off matrices363
Figure 13.1.2 A general pay–off matrix.363
Types of decision problem363
Decision–making under certainty363
Example 13.1.1364
Figure 13.1.3 Pay–off matrix for Example 13.1.1.364
Decisions under uncertainty364
Example 13.1.2364
Figure 13.1.4 Pay–off matrix for Example 13.1.2 (all figures are multiples of £1,000).365
Decisions under risk365
Example 13.1.3365
Table 13.1.1 Daily profits over 50 days.365
Figure 13.1.5 Pay–off matrix and probabilities for Example 13.1.3.365
Constrained decisions366
Sequential decision problems366
Figure 13.1.6 Part of a student’s decision tree.366
Exercises 13.1367
13.2 Decisions under uncertainty367
Decision criteria367
The Laplace decision criterion367
Example 13.2.1367
Table 13.2.1 Daily profits for day trading.368
Table 13.2.2 Average pay–offs for Example 13.2.1.368
The Wald (maximin) criterion368
Example 13.2.2368
Table 13.2.3 Row minima for Table 13.2.1.368
The Savage (minimax regret) decision criterion369
Example 13.2.3369
Table 13.2.4 Column maxima for Table 13.2.1.369
Table 13.2.5 Regret matrix for Table 13.2.4.369
Which criterion?369
Exercises 13.2370
13.3 Decisions under risk370
Incorporating probabilities371
Expected values371
Example 13.3.1371
Table 13.3.1 Estimated annual returns (£) for Example 13.3.1.371
Sequential decisions under risk372
Example 13.3.2372
Figure 13.3.1 Initial decision tree for Example 13.3.2.373
Table 13.3.2 Expected values for Example 13.3.2.373
Figure 13.3.2 Complete decision tree for Example 13.3.2.373
Misleading expected values374
Example 13.3.3374
Table 13.3.3 Pay–off matrix for Example 13.3.3.374
Other decision criteria and utility theory*375
Figure 13.3.3 A typical utility curve.375
Exercises 13.3376
13.4 Chapter review377
Chapter 14 An introduction to linear programming378
Introduction378
Chapter objectives378
14.1 Formulating decision problems as linear programs379
Linear programming379
Figure 14.1.1 Linear programming methodology.380
Formulating linear programs380
A basic standard form for linear programs380
Example 14.1.1381
Example 14.1.2382
Slack and surplus variables382
Example 14.1.3383
The linear equations384
General notation for linear programs*385
Exercises 14.1386
14.2 Solving small linear programs387
The graphical method387
The feasible region387
Example 14.2.1387
Figure 14.2.1 Non-negativity of two decision variables.388
Figure 14.2.2 The feasible region for Example 14.1.1.388
The objective function389
Example 14.2.2389
Figure 14.2.3 The objective function for different values of Z.389
The vertices of the feasible region390
Example 14.2.3390
Table 14.2.1 Evaluating the objective function at the vertices.390
Types of feasible region391
Example 14.2.4391
Figure 14.2.4 Graphical solution of Example 14.1.2 (in 1,000s acres).391
Exercises 14.2392
14.3 Solving large linear programs393
A note on installing Solver393
Entering a linear program into Excel – a very basic approach393
Example 14.3.1394
Figure 14.3.1 Entering a linear program into Excel.394
Using Excel Solver – a very basic approach394
Example 14.3.2396
Figure 14.3.2 Applying steps 1 to 8 of the basic Solver approach.397
Figure 14.3.3 Applying step 9 of the basic Solver approach.397
Using Excel Solver – a more general approach398
Figure 14.3.4 A more general approach for using Solver.398
Generating reports from Solver398
Figure 14.3.5 A solver answer report for Example 14.1.1.399
Figure 14.3.6 A solver sensitivity report for Example 14.1.1.399
Exercises 14.3400
14.4 Chapter review400
Chapter 15 An introduction to non-linear and other decision models401
Introduction401
Chapter objectives401
15.1 Decisions in business economics and operations management402
Optimising non-linear objectives402
Example 15.1.1402
Figure 15.1.1 Cost, revenue and profit for Example 15.1.1.403
Example 15.1.2403
An inventory problem404
Example 15.1.3404
Figure 15.1.2 Graph of stock over time.405
Figure 15.1.3 Total annual cost of stocks.406
Constrained non-linear problems406
Example 15.1.4407
Figure 15.1.4 Total annual cost of stocks with a capacity constraint.407
Example 15.1.5407
A general non-linear program and the Lagrangian*408
Example 15.1.6408
Table 15.1.1 Data for a three-product inventory problem.408
Table 15.1.2 Calculating the optimum values for the decision variables in Example 15.1.6410
Exercises 15.1410
15.2 Quality management decisions411
Control charts412
Figure 15.2.1 A process that appears to be in control.413
Figure 15.2.2 A process possibly out of control.413
Figure 15.2.3 A process out of control.413
Using control charts to support decisions413
Calculating the centre line and the control limtextts414
Figure 15.2.4 Six Sigma area under the Standard Normal distribution.415
Figure 15.2.5 Areas under a bell-shaped distribution within different control limtextts.415
Example 15.2.1415
Figure 15.2.6 A control chart for Example 15.2.1.416
Types of control chart416
Control charts for small samples of continuous variables417
Table 15.2.1 Control chart factors.418
Example 15.2.2418
Table 15.2.2 Service times for call centre in Example 15.2.2.419
Example 15.2.3419
Table 15.2.3 Service times for call centre in Example 15.2.3.420
Figure 15.2.7 R-Chart for the data in Examples 15.2.2 and 15.2.3.420
Figure 15.2.8 X¯-Chart for the data in Examples 15.2.2 and 15.2.3.420
Exercises 15.2421
15.3 Investment decisions422
Example 15.3.1422
Cash flow, inflow and outflow422
Figure 15.3.1 The cash flow for Example 15.3.1.422
Future values422
Example 15.3.2423
Figure 15.3.2 Future values for machine investment after 5 years.423
Table 15.3.1 Future values for Example 15.3.2423
Discounted Cash Flow (DCF) and Net Present Value (NPV)424
Example 15.3.3424
Figure 15.3.3 Discounted cash flow for Example 15.3.3.424
Calculating Net Present Value425
Internal Rate of Return (IRR)425
Calculating the Internal Rate of Return426
Example 15.3.4426
Table 15.3.2 Method of bisection for Example 15.3.4.426
Figure 15.3.4 Graph of NPV against discounting rate for Example 15.3.4.427
Payback period (PB)427
Calculating the payback period427
Example 15.3.5427
Table 15.3.3 Calculating the payback period at 20% discounting.428
Table 15.3.4 Calculating the payback period with no discounting.428
Figure 15.3.5 NPV over time at 20% discounting.428
Figure 15.3.6 NPV over time with no discounting.429
Exercises 15.3429
15.4 Chapter review430
Chapter 16 An introduction to decision problem network models431
Introduction431
Chapter objectives431
16.1 Networks and network diagrams432
Network optimisation problems432
Nodes, arcs and conventions432
Figure 16.1.1 Nodes connected by directed and undirected arcs.433
Network types433
Figure 16.1.2 Cyclic and acyclic networks.433
Arc values [capacities]433
Example 16.1.1433
Matrix representation434
Example 16.1.2434
Node values [capacities]434
Source and target nodes434
Paths and chains435
Example 16.1.3435
Figure 16.1.3 Bus routes for Example 16.1.3.435
Exercises 16.1436
16.2 Communication and distribution decisions437
The minimum spanning tree problem (MSTP)437
Example 16.2.1437
Figure 16.2.1 The MSTP network for Example 16.2.1.437
Figure 16.2.2 A tree for Example 16.2.1.438
Figure 16.2.3 A tree for Example 16.2.1.438
Prim’s algorithm for the MSTP438
Algorithm A4 (Prim)439
Example 16.2.2439
Laying out the MSTP calculations in a matrix440
Example 16.2.3440
The shortest path problem and Dijkstra’s algorithm441
Algorithm A5442
Example 16.2.4442
Exercises 16.2445
16.3 Project planning decisions446
Projects and project scheduling446
A methodology446
Drawing project networks447
Example 16.3.1447
Figure 16.3.1 A project network for Example 16.3.1.449
Simplifying project network diagrams449
Example 16.3.2450
Figure 16.3.2 A reduced project network for Example 16.3.1.450
Figure 16.3.3 A project network for Example 16.3.1 with no dummy arcs.450
Example 16.3.3450
Figure 16.3.4 Initial network for Example 16.3.3.451
Figure 16.3.5 Revised network for Example 16.3.3.451
Figure 16.3.6 A simplified network diagram for Example 16.3.2.451
Critical activities and the critical path452
The forward pass and earliest times452
Figure 16.3.7 An extended node notation.452
Example 16.3.4452
Figure 16.3.8 Initial network for the forward pass.453
Figure 16.3.9 Calculating the early time at node 2.453
Table 16.3.1 Calculations for earliest times at each node.454
Figure 16.3.10 The completed forward pass for Example 16.3.4.454
The backward pass and latest times454
Example 16.3.5455
Table 16.3.2 Calculations for latest times at each node.455
Figure 16.3.11 The earliest and latest times at each node.455
Critical path analysis455
Example 16.3.6456
Table 16.3.3 The results from a critical path analysis of Example 16.3.3.456
Exercises 16.3456
16.4 Chapter review458
Part C Closing comments459
Case C1 Firmecon460
Case C2 Pandbe Co.461
Case C3 Makit Ltd461
Case C4 PrintsRus462
Back Matter463
Further reading463
Table 1 Binomial distribution465
Table 2 Chi–squared distribution479
Table 3 Poisson distribution481
Table 4 Normal distribution493
Table 5 Student's t–distribution497
Index499

Book details

Vendor McGraw-Hill UK
SKU 9780077118822
ISBN-13 9780077118822
Author Frank Dewhurst
Edition 2nd
Category Business & Economics
Subject Statistics

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The new edition of Quantitative Methods for Business and Management offers a complete introductory course in Quantitative Methods, providing students with basic practical experience in quantitative approaches in modelling and analysis for business and management. The book features sections on foundation topics, models for business and management, and modelling and analyzing decisions. In particular, the new edition features greater coverage of statistics to reflect teaching in this area, with chapters on Elementary Statistics, Summary Statistics and Inferential Statistics. Other new areas of coverage in the second edition include Network Models and Non-linear Models. The book retains its popular style which offers students numerous examples accompanied by clear and straightforward explanations. Excel examples are also integrated throughout to help students to understand how this software tool is used by managers, and frequent questions and exercises enable students to test their understanding. A free CD contains Excel applications and solutions to the exercises in the textbook, and a full online learning centre completes an excellent learning package for business students.

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