Long Contents List of tables and figures Preface to the seventh edition Guided tour of the book Notes on the seventh edition Acknowledgements
I
xvi xxii xxiv xxvi xxvii
MATHEMATICAL APPLICATIONS
1
1 Revision mathematics Introduction Number and number operations Decimals, fractions and percentages Powers and roots Elementary algebra Indices and logs Mathematical symbols Graphs and straight lines Solving linear equations graphically Solving linear equations simultaneously Inequations Further reading
2 Keeping up with change: Index numbers Introduction Quantitative methods in action: Is the demand for oil increasing? Simple indices Weighted aggregate indices Consumer price indices Reflection Key points Further reading Practice questions Assignment
2 2 3 3 5 6 8 9 10 11 12 12 13
14 14 15 16 18 21 23 24 24 24 27
ix
x
LONG CONTENTS
II
COLLECTING AND INTERPRETING DATA
3 Collecting data: Surveys and samples Introduction Quantitative methods in action: The 2016 EU Referendum The basics of sampling Questionnaire design Simple random sampling Stratified sampling Multi-stage sampling Cluster sampling Systematic sampling Quota sampling Other sampling methods Obtaining research data from on-line sources Reflection Key points Further reading Practice questions Assignment
4 Finding patterns in data: Charts and tables Introduction Quantitative methods in action: The growth of China’s economy Data classification Tabulation of data Diagrammatic representation of data Creating charts with Excel 2019 Creating charts with IBM SPSS statistics Reflection Key points Further reading Practice questions Assignment
5 Making sense of data: Averages and measures of spread Introduction Quantitative methods in action: Are Australians better off than their UK counterparts? Measures of location Measures of spread Coefficient of variation Box and whisker plots
29 30 30 31 32 34 36 38 39 40 40 41 41 42 43 44 45 45 47
48 48 49 51 52 57 67 70 77 78 78 78 83
84 84 85 88 96 102 103
Long Contents
III
xi
Using Excel to calculate summary statistics Using SPSS to calculate summary statistics Reflection Key points Further reading Practice questions Assignment
105 107 111 111 111 112 115
PROBABILITY AND STATISTICS
117
6 Taking a chance: Probability Introduction Quantitative methods in action: Should you buy an extended warranty? Basic ideas The probability of compound events Conditional probability Permutations and combinations Expected value Reflection Further reading Practice questions Assignment
7 The shape of data: Probability distributions
118 118 119 121 122 125 132 133 134 135 135 137
138
Introduction Quantitative methods in action: Is goal scoring a random event? Discrete and continuous probability distributions The binomial distribution The Poisson distribution The normal distribution Reflection Key points Further reading Practice questions Assignment
138 139 140 140 147 151 161 162 162 162 165
8 Interpreting with confidence: Analysis of sample data
166
Introduction Quantitative methods in action: Using statistics to identify benefit fraud Samples and sampling Point estimates Sampling distribution of the mean
166 167 169 170 171
xii
LONG CONTENTS
Confidence intervals for a population mean for large samples Confidence intervals for a population mean for small samples Confidence interval of a percentage Calculation of sample size Finite populations Reflection Key points Further reading Practice questions Assignment
9 Checking ideas: Testing a hypothesis Introduction Quantitative methods in action: Are waiting times at A&E getting worse? The purpose of hypothesis testing Large sample test for a population mean Small sample test for a population mean The Z-test for a population percentage Hypothesis tests involving two population means Hypothesis tests involving two population percentages The chi-square hypothesis test SPSS and the chi-square test Reflection Key points Further reading Practice questions Assignment
10 Cause and effect: Correlation and regression Introduction Quantitative methods in action: Can you predict the number of medals won in an Olympic Games? Scatter diagrams Correlation Linear regression Coefficient of determination Using Excel to analyse bivariate data SPSS and scatter diagrams SPSS and linear regression Further topics in regression Reflection
174 176 178 179 180 181 181 182 182 183
184 184 185 187 188 192 193 195 202 203 210 213 214 214 215 218
219 219 220 223 225 230 233 233 236 237 238 245
Long Contents
IV
xiii
Key points Further reading Practice questions Assignment
245 246 246 249
DECISION-MAKING TECHNIQUES
251
11 How to make good decisions Introduction Quantitative methods in action: Multi-criteria decision analysis for student selection in a Brazilian university Problem formulation Payoff tables Decision trees Utility Multi-criteria decision-making (MCDA) Reflection Key points Further reading Practice questions Assignment
12 Choosing wisely: Investment appraisal Introduction Quantitative methods in action: Insuring Against Business Interruption Risk Measures of investment worth Traditional methods for comparing projects Discounted cash flow techniques Other applications of the compound interest formula Reflection Key points Further reading Practice questions Assignment
13 Forecasting: Time series analysis Introduction Quantitative methods in action: Global warming and its effect on the occurrence of salmonella in Singapore The decomposition model Isolating the trend
252 252 253 255 260 264 273 276 284 284 284 285 290
291 291 292 294 295 296 303 307 308 308 308 310
312 312 313 315 317
xiv
LONG CONTENTS
Isolating the seasonal component Analysis of errors Seasonally adjusted series Forecasting using the decomposition model Exponential smoothing Using spreadsheets in time series analysis Reflection Key points Further reading Practice questions Assignment
14 Making the most of things: Linear programming Introduction Quantitative methods in action: Generating a schedule for sports leagues with reference to the Australian Football League Basics of linear programming Model formulation Graphical solution of linear programming problems Tight and slack constraints Sensitivity analysis Minimization problems Using Excel’s ‘Solver’ to solve linear programming problems Applications of linear programming Reflection Key points Further reading Practice questions Assignment
15 Planning large projects: Network analysis Introduction Quantitative methods in action: London’s Crossrail project The activity-on-node method The float of an activity Resource scheduling Cost scheduling Handling uncertainty in time estimates (the PERT method) Reflection Key points Further reading
320 323 326 327 328 330 333 333 334 334 337
338 338 339 341 341 343 346 346 349 351 356 366 367 367 367 373
375 375 376 377 380 380 383 386 389 389 390
Long Contents
Practice questions Assignment
xv
390 396
16 Managing stock levels: Materials management and inventory control
398
Introduction Quantitative methods in action: Inventory management at Zara Costs of holding stock Economic Order Quantity (EOQ) model Discounts Uncertainty in demand Reflection Key points Further reading Practice questions Assignment
398 399 400 400 404 405 407 407 407 408 409
Bibliography Appendix: statistical tables Index
410 411 423
1
Revision mathematics INTRODUCTION The aim of this chapter is to provide the basic numeracy skills that will be needed in subsequent chapters. If you have any doubts about your knowledge of particular areas you are recommended to tackle the diagnostic tests that you will find throughout this chapter. If you do not achieve 100% success in any of these tests, try to find out where you went wrong by reading through the remainder of that section. The books listed in the further reading section may also be of help if you need additional practice.
2
1
Revision mathematics
3
Number and number operations activity
1.1
Write down the answers to the following questions (without a calculator). 1
4 1 (22)
2
4 2 (24)
3
5 1 (22) 2 (24)
4
2 3 (23)
5
215 4 (23)
6
210 2
7
5 0
8
0 5
9 3 3 218 4 4
10
3 3 (2 1 8) 4 4
Answers
1 2 4 26 7 no solution (`) 10 7.5
2 8 5 5 8 0
3 7 6 25 9 8
Number operations often confuse people, particularly when they involve positive and negative numbers. The easy rule is that like signs give a plus while unlike signs give a negative. This is summarized in Table 1.1. Another problem is deciding the order in which to carry out a series of calculations. Questions 9 and 10 had the same numbers in the same order but in question 10 a bracket separated some of the figures. There is an order in which mathematical operations are performed. You may have come across the term ‘bedmas’, which stands for ‘brackets, exponent, division, multiplication, addition and subtraction’. This is not strictly correct as addition and subtraction have equal priority, as do multiplication and division. Where the operations have equal priority the order is taken from left to right. The main point to remember is that if you want a calculation to be done in a particular order you should use brackets to make the order clear. Most scientific calculators should use the bedmas system but it is worth checking with some simple calculations. These calculators will also have the bracket function to allow the order of calculation to be controlled. Table 1.1 Number operations Addition and subtraction
Multiplication
Division
1(1)5 1
1 3 151
1 4 151
2(2)5 1
2 3 251
2 4 251
1(2)5 2
1 3 252
1 4 252
2(1)5 2
2 3 152
2 4 152
Decimals, fractions and percentages activity
1.2
1 Convert the following decimals to fractions in their simplest form. # (a) 0.2 (b) 0.5 (c) 0.333 (d) 0.125 (e) 0.375
1
4
I
MATHEMATICAL APPLICATIONS
2 Convert the following fractions to decimals. 1 1 11 2 3 (b) (c) (d) (e) (a) 10 5 3 7 8 7 1 3 (g) (h) 1 (f) 3 100 4 3 Convert the following fractions to percentages. 1 1 1 2 3 (b) (c) (d) (e) (a) 4 2 5 3 4 4 Convert the following percentages into fractions in their simplest form. (a) 40% (b) 45% (c) 60% (d) 12% (e) 19% 5 Carry out the following calculations. 1 1 (b) 3 3 (c) of 46% (a) 0.25 1 0.37 2 2 3 3 3 3 4 2 (e) 3 (f) 4 (d) 1 5 3 4 2 4 2 1 (g) 0.25 3 (h) 0.00025 3 3000 5 6 Round the following decimals to 3 decimal places. (a) 1.5432 (b) 1.5438 (c) 1.5435 (d) 0.000843 (e) 100.2003 7 Write the following numbers to 3 significant figures. (a) 1.5432 (b) 1.5438 (c) 1.5435 (d) 0.000843 (e) 100.2003 (f) 13 256 (g) 1.000561 8 Write down the following numbers in scientific notation. (a) 25 438 176 (b) 1 600 000 (c) 0.00001776 Answers
1 5 2 (a) 0.1 # (f) 2.333
1 2 (b) 0.2 (g) 0.01
1 3 # (c) 3.666 (h) 1.75
3 (a) 25% 2 4 (a) 5
(b) 50% 9 (b) 20
(c) 20% 3 (c) 5
5 (a) 0.62
(b) 1.5
(c) 23%
1 2 6 (a) 1.543
1 or 0.05 2 (b) 1.544
7 (a) 1.54 (f) 13 300
(b) 1.54 (g) 1.00
1 (a)
(f)
(b)
(g)
8 (a) 2.5438176 3 107
(c)
1 8 (d) 0.2857
3 8 (e) 0.375
(d)
(e)
# (d) 66.666% 3 (d) 25 7 (d) 1 15
(e) 75% 19 (e) 100 1 (e) 1 8
(c) 1.544
(d) 0.001
(e) 100.200
(c) 1.54
(d) 0.000843
(e) 100
(h) 0.75
(b) 1.6 3 106
(c) 1.776 3 1025
Fractions, decimals and percentages are used throughout this book and you need to be able to convert a number from one form to another. To convert from a fraction to either a
1
5
Revision mathematics
decimal or a percentage all you need to do is to divide the numerator (the number above the line) by the denominator (the number below the line). For a percentage you also need to multiply by 100. To convert a decimal to a fraction you simply have to remember that the first digit after the decimal point is a tenth and the second digit is a hundredth and so 2 and this can be simplified by dividing the top and bottom by 2 on. For example, 0.2 is 10 1 to give . The conversion of a percentage to a fraction is easier as the denominator of the 5 2 40 and this simplifies to . fraction is always 100. So 40% is 100 5 A frequent question asked by students is, how many decimal places should an answer be given to? There is no easy answer to this, although a good rule of thumb to use when analyzing data is to give one more place than in the original data. So if your data had one decimal place you should give your answer to two decimal places. Giving more decimal places implies that your answer is accurate to this number of decimals, which is misleading. However, numbers should only be rounded at the end of a calculation. If you round intermediate values you will introduce rounding errors into your calculations, which can cause major errors in your final figure. When rounding, remember the rule that numbers below 5 are rounded down and numbers of 5 and above are rounded up. So 1.5432 is 1.543 to 3 decimal places but 1.5435 is 1.544 to 3 decimal places. Another form of cutting down the number of digits in an answer is to give a stated number of significant figures. In many cases the two systems will give identical results, but the real difference with using significant figures is that zeros are only included when they are significant. Thus 0.01654 becomes 0.0165 to three significant figures while 1.01654 would become 1.02. Significant figures can also be used when handling large numbers. For example, 13 256 is 13 000 to 2 significant figures. The zeros have to be added to maintain the place value of the remaining numbers. If very large or very small numbers are involved it is all too easy to make a mistake and accidentally lose a digit. To avoid this you can use scientific notation. Scientific notation is in the form of a 310b where a is a decimal number below 10 and b is the number of places the decimal point needs to be moved to arrive back at the original number. For example, 25 438 176 can be changed to 2.5438176 3107 as the decimal place has been moved 7 places to the left. Scientific calculators use this method when the number exceeds a set number of digits, except that the base 10 is not displayed – a source of confusion among students!
Powers and roots activity
1.3
Write down the value of the following. 1
23
2
34
3
1 3 a b 2
4
223
5
"81
6
3 " 27
2
81
3
1 8
4
1 8
5
9
6
3
Answers
1
8
1
6
I
MATHEMATICAL APPLICATIONS
When a number is raised to a power (or exponent) the number is multiplied by itself that number of times. So 23 means 2 3 2 3 2 5 8. Powers take precedence in an expression after any brackets. In the calculation of 2 1 6 3 32 2 4, the order of calculation is as follows. 32 5 9, 9 3 6 5 54, 54 1 2 2 4 5 52
Elementary algebra activity
1.4
1 Simplify the following expressions. (a) x 1 x2 1 y 1 2x 1 3x2
(b)
2x x
(c)
2x y
(d)
x3 x2
x2 2 4 (x 1 2) 2 Evaluate the expressions given in question 1 when x 5 2 and y 5 4. 3 Solve the following equations. 7x 2 4 5 2x 1 4 (c) x2 1 4 5 20 (b) (a) 4x 2 5 5 9 2 1 (d) 5 10 (f) 2x2 2 6x 2 3 5 0 (e) x2 2 4x 1 3 5 0 x (e)
Answers
2x y
1 (a) 4x2 1 3x 1 y
(b) 2
(c)
2 (a) 26 3 (a) 3.5 (f) −0.436 or 3.436
(b) 2 (b) 4
(c) 1 (c) 64
(d) x
(e) x 2 2
(d) 2 (d) 0.1
(e) 0 (e) 1 or 3
Most students who have not studied mathematics beyond GCSE level find algebra difficult. The idea of replacing numbers by letters is quite abstract and many students do not understand the need for it. Unfortunately some understanding of algebra is essential if you are to successfully complete a quantitative methods course and many chapters in this book assume this understanding. There are many algebraic techniques but the most important as far as this book is concerned is to be able to solve equations. To achieve this it is necessary to remember that whatever you do to one side of the equation you must do to the other. In solving 7x 2 4 5 2x 1 4 2 the following steps are carried out: 1 2 3 4
Multiply both sides by 2. The equation then becomes 7x 2 4 5 4x 1 8. Subtract 4x from both sides to give 3x 2 4 5 8. Add 4 to both sides to give 3x 5 12. Divide both sides by 3.
1
Revision mathematics
7
This gives the answer x 5 4. When the unknown variable is raised to a power, the method of solving the equation becomes more complicated and in some cases it is necessary to resort to numerical methods. Quadratic equations are those involving x2 and the method of solution depends on the equation. For x2 1 4 5 20 the solution follows the method already described. That is, subtract 4 from both sides x2 5 16 and then take the square root of both sides x 5 "16 5 64 Notice that both 14 and 24 are solutions since (24) 2 5 16. Some quadratic equations can be factorized. To factorize an expression we need to find two factors that when multiplied together give the original expression. x2 1 2x 5 0 can be factorized by noting that x is a common factor to both terms. If we take out the x we get x(x 1 2) 5 0 and both x and (x 1 2) are factors of the original expression. The modified equation is easier to solve than the original as we can see that the equation is true when either x 5 0 or (x 1 2) 5 0. That is, x 5 0 or x 5 22. For x2 2 4x 1 3 5 0 we need to find two brackets that when multiplied together give these terms. Factorizing this type of expression can involve some trial and error but in this case it is fairly easy and will be (x 2 3) (x 2 1) 5 0 If you multiply the two brackets you get x2 2 3x 2 x 1 3 5 0 which is the same as the original since 23x 2 x 5 24x. The solution to this equation is either x 2 3 5 0 or x 2 1 5 0 So x is either 1 or 3. Factorizing only works easily for a small number of quadratic expressions, and a more general method is to use a formula. If the expression is in the form ax2 1 bx 1 c 5 0, the formula is: x5
2b 6 "b2 2 4ac 2a
For example, to solve 2x2 2 6x 2 3 5 0, we first note that a 5 2, b 5 26 and c 5 23 so x 5 5
2 (26) 6 "(26) 2 2 4 3 2 3 (23) 232 6 6 "36 1 24 4
6 6 7.74597 4 5 20.436 or 3.436
5
1
8
I
MATHEMATICAL APPLICATIONS
Indices and logs activity
1.5
1 Simplify the following expressions. (b) 32 3 33 (a) 23 3 25 105 (d) (e) (23 ) 2 102 ax (g) b5 3 c5 (h) y a a2x (j) (k) 163/2 a5x 2 Solve for x: (b) ex 5 1.6 (a) 10x 5 15 5x 3 (d) 1.8 5 x (e) 52x13 5 12 3 64x 3
(c) 23 3 84 (f) ax 3 a2x (i) (ax ) y
(c) 25 5 10 3 (1.1) x
Answers
1 (a) 28 5 256 (e) 26 5 64 (i) axy 2 (a) 1.176
(b) 35 5 243 (f) a3x (j) a23x (b) 0.470 (c)
(c) 215 5 32768 (d) 103 5 1000 (g) (bc) 5 (h) ax2y (k) 64 9.614 (d) 0.1338 (e) 0.5935
In Section 1.4 we saw that if a number is multiplied by itself a number of times then we can simplify the expression by the use of indices – so that 2 3 2 3 2 5 23. In general the product of a 3 a 3 a . . . is written as an. We say that the base (a) is raised to the power of n. There are 3 important formulae for indices and these are: am 3 an 5 am1n am 4 an 5 am2n (am ) n 5 amn Therefore, providing the bases are the same, you can add or subtract the powers. So 23 3 25 5 2315 5 28 5 256 and
105 5 10522 102 5 103 5 1000
If m and n have the same value (say x) then ax 4 ax 5 a0. As this expression must also equal 1, it follows that a0 5 1. Also a1 5 a. The power can be negative or fractional. 1/an can also be written as a2n, and a1/2 is the same as "a. We could simplify 163/2 to ( (16) 1/2 ) 3. Since the square root of 16 is 4, then 43 5 64.
1
Revision mathematics
9
We can rewrite Y 5 ax as x 5 log aY. The word log is short for logarithm and this expression is read as x equals log of Y to the base a. There are only two bases you need worry about; these are base 10 and base e. These bases can be found on all scientific calculators; log to base 10 is usually written as log, and log to base e is written as ln. There are three important formulae for logs that apply to all bases and these are: log (uv) 5 log u 1 log v log (u/v) 5 log u 2 log v log xn 5 n log x
1
The importance of logs as far as this book is concerned is the ability to use the laws of logs to solve certain types of equations. For example, to solve 52x13 5 12 3 64x you would carry out the following steps. log(52x13 ) 5 log(12 3 64x ) (2x 1 3) log 5 5 log 12 1 log 64x 2x log 5 1 3 log 5 5 log 12 1 4x log 6 2x log 5 2 4x log 6 5 log 12 2 3 log 5 2x(log 5 2 2 log 6) 5 log 12 2 3 log 5 log 12 2 3 log 5 2x 5 log 5 2 2 log 6 1.0792 2 3 3 0.6990 2x 5 0.6990 2 2 3 0.7782 21.0178 2x 5 20.8574 2x 5 1.1871 x 5 0.5935
Mathematical symbols activity
1.6
What do the following symbols stand for? 1
,
2
#
3
.
4
$
5
2
6
>
7
a
Answers
1 3 5 7
less than greater than not equal to the sum of
2 less than or equal to 4 greater than or equal to 6 approximately equal to
We will be using all these symbols in later chapters of the book, particularly the summation sign. If you see the expression a x it means you should add up all the values of x, while a xy means multiply the pairs of x and y before summing them. For example, in Table 1.2 there are two columns of numbers which I have called x and y.
10
I
MATHEMATICAL APPLICATIONS
From Table 1.2
ax5211141614
5 17
a xy 5 14 1 5 1 8 1 18 1 16 5 61 2 a x 5 4 1 1 1 16 1 36 1 16 5 73
If your calculator has the SD or Statistical Data function many of these calculations can be done automatically for you. Table 1.2 Use of the summation sign x
y
xy
2 1 4 6 4
7 5 2 3 4
14 5 8 18 16
Graphs and straight lines activity
1.7
Which of the following equations represent straight lines? 1
y 5 2x 1 5
2
y 1 4x 5 20
3
y58
4
y5x
5
y 5 x2
Answers
All except equation 5. The equation of a straight line is y 5 mx 1 c, where m is the gradient and c is the value of y when x 5 0 (often referred to as the intercept on the y-axis). Equation 1 is obviously a straight line with m 5 2 and c 5 5. Equation 2 is also a straight line and this may be easier to see if you subtract 4x from both sides of the equation. That is: y 5 20 2 4x or
y 5 24x 1 20
This equation is now in the standard form and you should see that m 5 24 and c 5 20. Equation 3 is another straight line but this is a special one. This could be written as y 5 0x 1 8 and you will see that m 5 0; that is, the gradient is zero. This can be therefore be represented as a horizontal line. Equation 4 can be written as y 5 x 1 0, so this represents a straight line with a gradient of 1 and c of 0. This line passes through the origin of the graph (that is, x 5 0, y 5 0). Equation 5 is not a straight line since y increases as the square of x. The graphs of these equations are shown in Figure 1.1.
1
Revision mathematics
11
28 26
y = x2
24 22 y + 4x = 20
20 18
y = 2x + 5
16 y 14
1
12 10
y=8
8 6 4 2 0
0
2
4
x
Figure 1.1 Graph for Activity 1.7
Solving linear equations graphically activity
1.8
Plot the equations 2y 1 x 5 8 and y 1 2x 5 7 on the same graph and write down the coordinates of the point of intersection. 10 9 8 7
y + 2x = 7
6 y 5 4 3 2
2y + x = 8
1 0
0
2
4 x
6
8
Figure 1.2 Solving equations graphically
You should have obtained the graph shown in Figure 1.2. The point of intersection of the two lines is at the point x 5 2, y 5 3. This can be written as (2,3). In this example the
12
I
MATHEMATICAL APPLICATIONS
coordinates of the point of intersection were both whole numbers, which made it easy to read from your graph. Unfortunately this is not always the case and accurately reading fractional values from a graph is difficult. A better method is the algebraic method of simultaneous equations.
Solving linear equations simultaneously activity
1.9
Solve the equations given in Activity 1.8 simultaneously. All being well, your answer agreed with the graphical method. The method of solving simultaneous equations I prefer is the method of elimination. To use this method the two equations are written down as follows: y 1 2x 5 7
(1)
2y 1 x 5 8
(2)
The coefficients of either x or y must be equal in the two equations and this is achieved by multiplying both sides of equation (1) by 2. The equations can now be subtracted: 2y 1 4x 5 14 2y 1 x 5 8 3x 5 6 (y has now been ‘eliminated’) That is
x52
This value of x can now be substituted back into either equation (1) or (2) to give the value of y. Using equation (2): 2y 1 2 5 8 2y 5 6 Therefore
y53
Inequations activity
1.10
1 What do the following inequations mean? (a) y 1 2x # 7 (b) 2y 1 x $ 8 2 How would you represent these two inequations graphically? And what does the term ‘intersection’ mean in this case? Answers
1 Inequation (a) means that the sum of the left hand side of the equation must be less than or equal to 7; that is, the sum cannot be greater than 7. Inequation (b) means that the sum of the left hand side of the equation must be at least 8; it cannot be less.
1
Revision mathematics
13
2 Whereas an equation can be represented by a straight line, an inequation is represented by a region. The equation y 1 2x 5 7 forms the boundary of the region represented by the inequation y 1 2x # 7. Similarly the equation 2y 1 x 5 8 forms the boundary of the region 2y 1 x $ 8. The region can only be on one side of the boundary and this can be found by inspection; that is, a point is investigated to see if it satisfies the inequation. The easiest point to try is the origin (x 5 0, y 5 0), except when the boundary passes through this point. When the region has been found it needs to be identified. This can be done by shading. The normal convention is to shade the unwanted region and this is the convention adopted in this book. You can see this in Figure 1.3. The intersection of regions is the area on the graph that satisfies all inequations. In this example, the area is represented by ABC. Any point within this area (including the boundaries) satisfies both inequations. 10 9 8 7
B
6 y 5
This area satisfies both inequations
A
4
C
3 2 1 0
0
2
4 x
6
8
Figure 1.3 Graph for Activity 1.10
Further reading
There are a number of texts that are designed to help the student with basic mathematics such as Rowe (2002) and Morris (2007). There are also many online sources such as the BBC’s Skillswise (www.bbc.co.uk/skillswise/maths) which is part of their online learning support series.
1
Index χ2 distribution see Chi-square distribution χ2 hypothesis test see Chi-square hypothesis test A a priori approach to probability 122 accounting rate of return 295–6 activity on node method 377–80 addition rule of probability 124 additive time series model 320–3 ANOVA 242 ARR see accounting rate of return averages 88–96 mean 88–95 median 88–95 mode 88–94 B bar charts 58–60 component 58–9 multiple 58–9 percentage 58–9 simple 58 using Excel 67–70 using SPSS 70–7 Bayes’ theorem 129–32, 267–8 binomial distribution 140–7 approximating by the normal distribution 159–6 approximating by the Poisson distribution mean and standard deviation of 146–7 using Excel 145 using tables 145 box and whisker plot 103–5 buffer stock 405 C categorical data 51 causal loop diagrams 258 central limit theorem 172 central tendency, measures of see averages centred moving averages 319–23 charts bar 58–60 box and whisker 103–5 frequency polygon 63 Gantt 381 histogram 61–3 line graph 60–1 ogive 64–6 pie 57
150
resource histogram 382 scatter diagrams 223–5 Stem and leaf plot 66–7 using Excel 67–70 using SPSS 72–7 chi-square hypothesis test 203–13 goodness-of-fit test 204–6 test of association 206–9 cluster sampling 40 coefficient of correlation see correlation coefficient of determination 233 coefficient of variation 102 cognitive maps 256–7 collection of data 33 combinations and permutations 132–3 component bar chart 58–9 compound events 122 compound interest 296–8 conditional probability 125–6 confidence intervals of a mean 174–8 of a percentage 178–9 constant repayments 304–5 Consumer price index (CPI) 21–3 continuity correction 213 continuous compounding 306–7 continuous data 51 continuous probability distribution 140, 151–61 correlation 219–31 Pearson’s product moment correlation 228–30 scatter diagrams 223–5 Spearman’s rank correlation coefficient 226–8 using Excel 233–6 using SPSS 236–9 CPA see critical path analysis Cramer’s V statistic 211 crashing 383–6 critical path 380 critical path analysis activity-on-node 377–80 crashing 383–6 critical path 380 float 380 Gantt chart 381 PERT method 386–9 resource histogram 382 resource scheduling 380–3 crosstabs 210–11 cumulative frequency ogive see ogive
423
424
INDEX
D data categorical 51 collection of 33 continuous 51 discrete 51 grouped 55 interval scale 51 nominal 51 ordinal 51 presentation of 57–67 ratio scale 51–2 summarising 88–110 decision criterion expected monetary value (EMV) 261 expected opportunity loss 261 Hurwicz criterion 260 maximax 260 maximin 260 minimax regret rule 260 decision-making causal loop diagrams 258–9 cognitive maps 256–7 decision trees 264–70 efficient frontier 280 influence diagrams 259–60 mind maps 255 multi-criteria 276–83 payoff tables 260–3 rich picture diagrams 255–6 utility 273–6 value of imperfect information 270 value of perfect information 263 decision trees 264–73 and Bayes’ theorem 267–70 sensitivity analysis 270–3 using Excel 272–3 decomposition model see time series depreciation 306 diagrammatic representation of data 57–77 discount factor 298 discounted cash flow techniques 296–300 discrete data 51 discrete probability distributions 140–51 dispersion see spread, measures of E economic order quantity (EOQ) model efficient frontier 280 EMV see expected monetary value EOQ see economic order quantity Excel binomial calculations 145 charts 67–70 correlation and regression 233–6 decision trees 272–3 hypothesis tests 209–10 linear programming 351–6 Poisson calculations 148 Solver 351
400–3
summary statistics 88–110 time series 330–2 expected monetary value (EMV) 261 expected opportunity loss 261 expected value 133–4 exponential smoothing 328–30 F finite population correction factor 180 forecasting see Time series analysis frame, sampling 32 frequency polygon 63 G Gantt chart 381 Goal programming 361–6 goodness-of-fit test 204–6 H Hiview 3 281 histogram 61–3 Hurwicz criterion 260 hypothesis tests chi-square hypothesis tests 203–13 correlation coefficient 239–40 goodness-of-fit test 204–6 large sample test for a population mean 188–92 large sample test for two independent population means 195–6 paired t-test 199–202 population percentage 193–5 small sample test for a population mean small sample test for two independent population means 196–9 test of association 206–9 two population percentages 202–3 using Excel 209–10 using SPSS 198–9, 201–2, 210–13
192–3
I index numbers Consumer price index (CPI) 21–3 Laspeyres’ index 18–9 Paasche’s index 20–1 retail price index (RPI) 21–3 simple indices 16–8 weighted aggregate 18–23 inference see confidence intervals and hypothesis tests influence diagrams 259–60 internal rate of return (IRR) 300–2 interquartile range 96–7 interval estimates see confidence intervals inventory control 398–407 buffer stock 405 discounts 404–5 economic order quantity (EOQ) model 400–3 time between orders 403 uncertainty in demand 405–7 investment appraisal 291–307
Index
accounting rate of return (ARR) 295–6 compound interest 296–8 constant repayments 304–5 continuous compounding 306–7 depreciation 306 discount factor 298 discounted cash flow techniques 296–300 internal rate of return 300–2 net present value (NPV) 298–300 payback method 295 present value 298 simple interest 296 sinking funds 303–4 IRR see internal rate of return J judgemental sampling 41 L large sample test for a population mean 188–92 large sample test for two independent population means 195–6 Laspeyres’ index 18–9 least squares method 232–3 line graph 60–1 linear programming 338–366 Goal programming 361–6 graphical solution 343–6 minimisation problems 349–51 model formulation 341–2 sensitivity analysis 346–9 shadow price 347 tight and slack constraints 346 Transportation problems 356–61 using Excel (Solver) 351–6 linear regression 230–245 coefficient of determination (R2) 233 method of least squares 232–3 using Excel 233–6 using SPSS 236–9 location, measures of see averages LP see linear programming M maximax criterion 260 maximin criterion 260 mean 88–95 of a binomial distribution 146–7 of a normal distribution 151 of a Poisson distribution 151 mean absolute deviation 325–6 mean square error 325–6 median 88–95 method of least squares 232–3 mind maps 255 minimax regret rule 260 mode 88–94 moving averages 317–20 multi-criterion decision-making 276–83 efficient frontier 280 SMART 276–81
425
swing weights 279 Using Hiview 3 281 value tree 276–81 multiple bar chart 58–9 Multiple regression 241–5 multiplication rule of probability 125 multiplicative time series model 320–3 multi-stage sampling 39–40 N net present value (NPV) 298–300 Network analysis see Critical path analysis nominal data 51 non-probabilistic sampling 34 normal distribution 151–61 approximating a binomial distribution 159–60 approximating a Poisson distribution 161 standard normal distribution 153–9 O ogive 64–6 ordinal data 51 P p-value 198 Paasche’s index 20–1 paired t-test 199–202 payback method 295 payoff tables 260–3 Pearson’s product moment correlation coefficient 228–30 percentage bar chart 58–9 permutations and combinations 132–3 Phi statistic 211 pie chart 57 Point estimates 170 Poisson distribution 147–51 approximating by the normal distribution 161 approximating to a binomial distribution 151 mean and standard deviation of 151 using Excel 148 using tables 149 posterior probability 267 present value 298 presentation of data diagrammatic methods 57–77 using Excel 67–70 using SPSS 70–7 prior probability 267 probabilistic sampling 34 probability 118–134 a priori 122 addition rule 124 Bayes’ theorem 129–32 compound events 122 conditional 125–6 distributions 138–61 empirical 121 expected value 133–4 independent events 122
426
INDEX
multiplication rule 125 mutually exclusive events 122 mutually exhaustive outcomes 122–5 permutations and combinations 132–3 posterior 267 subjective 121 tree diagrams 126–8 Venn diagrams 123–4 probability distributions 138–61 binomial 140–7 continuous 140, 151–61 discrete 140–51 normal 151–61 Poisson 147–51 purposive sampling 41 Q questionnaire design quota sampling 41
34–6
R range 96 ratio scale data 51–2 regression see linear regression residuals see errors, analysis of time series resource histogram 382 resource scheduling 380–3 retail price index (RPI) 21–3 return on capital employed see accounting rate of return rich picture diagrams 255–6 S sampling frame 32 sampling methods 32–43 cluster 40 judgemental 41 multi-stage 39–40 non-probabilistic 34 probabilistic 34 purposive 41 quota 41 simple random 36–8 snowball 41 stratified 36–8 systematic 40 scatter diagrams 223–5 seasonal component 320–3 seasonally adjusted series 326–7 sensitivity analysis decision trees 270–3 linear programming 346–9 shadow price 347 significance testing see hypothesis testing simple bar chart 58 simple indices 16–8 simple interest 296 simple random sampling 36–8 sinking funds 303–4 small sample test for a population mean 192–3
small sample test for two independent population means 196–9 SMART 276–81 snowball sampling 41 Solver 351–6 Spearman’s rank correlation coefficient 226–8 spread, measures of 96–105 interquartile range 96–7 range 96 standard deviation 97–101 variance 97–101 SPSS charts 72–7 correlation and regression 236–9 crosstabs 210–1 hypothesis tests 198–9, 201–2, 209–13 summary statistics 107–110 standard deviation of a binomial distribution 146–7 of a frequency distribution 101–2 of a normal distribution 153–61 of a Poisson distribution 151–2 of a sample 97–101 standard error of a percentage 178, 194 of a regression 239 of the mean 173, 188 of the sampling distribution 173 of two independent means 196 of two population percentages 202 of paired samples 200 standard normal distribution 153–9 Stem and leaf plot 66–7 stock control see inventory control stratified sampling 36–8 subjective probability 121 summarising data 88–110 averages 88–96 spread 96–105 using Excel 88–110 using SPSS 107–110 survey methods see sampling methods systematic sampling 40 T t-test for a mean 192–3 test for a population mean 188–93 test for goodness-of-fit 204–6 test for two population means 195–9 test of a population percentage 193–5 test of association 206–9 test of two population percentages 202–3 tests of hypothesis see hypothesis tests time series analysis 312–333 additive model 320–3 analysis of errors 323–6 centred moving averages 319–23 decomposition model 315–28 exponential smoothing 328–30
Index
forecasting 327–8 mean absolute deviation 325–6 mean square error 325–6 moving averages 317–20 multiplicative model 320–3 seasonal component 320–3 seasonally adjusted series 326–7 trend 315, 317–20 using Excel 330–2 Transportation algorithm 356–61 U utility
273–6
427
V value of imperfect information 270 value of perfect information 263 variance 97–101 Venn diagram 123–4 W weighted aggregate index
18–23
Z Z test for a population percentage 193–5 Z test for two population percentages 202–3