CIMA Exam Practice Kit: Business Mathematics

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CIMA Exam Practice Kit

Business Mathematics

This Page is Intentionally Left Blank

CIMA Exam Practice Kit

Business Mathematics Walter Allan

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

CIMA Publishing An imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published 2005 Copyright © 2005, Elsevier Ltd. All rights reserved No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK: phone: (44) (0) 1865 843830; fax: (44) (0) 1865 853333; e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN

0 7506 6586 6

For information on all CIMA Publishing Publications visit our website at www.cimapublishing.com Typeset by Integra Software Services Pvt. Ltd, Pondicherry, India www.integra-india.com Printed and bound by Krips, Holland

Contents

About the Author Introduction Syllabus Guidance, Learning Objectives and Verbs Examination Techniques Mathematical Tables

vi vii viii xi xviii

1

Formulae

1

2

Equations and Graphs

7

3

Percentages, Ratios and Proportions

15

4

Accuracy and Rounding

20

5

Financial Mathematics

23

6

Data Collection

29

7

Presentation of Data

32

8

Averages

40

9

Variation

46

10

Index Numbers

53

11

Probability

59

12

Expected Value and Decision-Making

66

13

The Normal Distribution

75

14

Statistical Inference

82

15

Correlation and Regression

88

16

Time Series

95

17

Mock Examination

102

18

Mock Assessment

113

v

About the Author

Walter Allan is a graduate of Heriot-Watt University where he specialized in Mathematical Economics and probability theory. He has extensive lecturing experience in both the public and private sectors. He is responsible for the teaching of economics at Cass Business School, London and the Actuarial Science Degree.

vi

Introduction

Welcome to the new CIMA Exam Practice Kit which has been launched to coincide with a major change in the syllabus where new examinations will take place from May 2005. This Kit has been designed with the needs of home study and distance candidates in mind, and it is also ideal for fully taught courses or for students resitting papers from the old syllabus. These hints, question and answers have been produced by some of the best known freelance tutors in the United Kingdom who have specialized in their respective papers and the questions and topics selected are relevant for the May 2005 and November 2005 examinations. The exam practice kits will complement CIMA’s existing study manuals with the Q & A’s from May 2005 published in the next edition of the CIMA study manual and the Q & A’s from November 2005 examination published in the 2006 edition of the CIMA Exam Practice Kit. Good luck with your studies.

vii

Syllabus Guidance, Learning Objectives and Verbs

A The syllabus The syllabus for the CIMA Professional Chartered Management Accounting qualification 2005 comprises three learning pillars: 1 2 3

Management Accounting pillar Business Management pillar Financial Management pillar.

Within each learning pillar there are three syllabus subjects. Two of these subjects are set at the lower ‘Managerial’ level, with the third subject positioned at the higher ‘Strategic’ level. All subject examinations have a duration of 3 hours and the pass mark is 50%. Note: In addition to these nine examinations, students are required to gain three years relevant practical experience and successfully sit the Test of Professional Competence in Management Accounting (TOPCIMA).

B Aims of the syllabus The aims of the syllabus are: • To provide for the Institute, together with the practical experience requirements, an adequate basis for assuring society that those admitted to membership are competent to act as management accountants for entities, whether in manufacturing, commercial or service organisations, in the public or private sectors of the economy. • To enable the Institute to examine whether prospective members have an adequate knowledge, understanding and mastery of the stated body of knowledge and skills. • To complement the Institute’s practical experience and skills development requirements. viii

Syllabus Guidance, Learning Objectives and Verbs ix

C Study weightings A percentage weighting is shown against each topic in the syllabus. This is intended as a guide to the proportion of study time each topic requires. All topics in the syllabus must be studied, since any single examination question may examine more than one topic, or carry a higher proportion of marks than the percentage study time suggested. The weightings do not specify the number of marks that will be allocated to topics in the examination.

D Learning outcomes Each topic within the syllabus contains a list of learning outcomes, which should be read in conjunction with the knowledge content for the syllabus. A learning outcome has two main purposes: 1 2

to define the skill or ability that a well-prepared candidate should be able to exhibit in the examination; to demonstrate the approach likely to be taken by examiners in examination questions.

The learning outcomes are part of a hierarchy of learning objectives. The verbs used at the beginning of each learning outcome relate to a specific learning objective, for example, evaluate alternative approaches to budgeting. The verb ‘evaluate’ indicates a high level learning objective. As learning objectives are hierarchical, it is expected that at this level, students will have knowledge of different budgeting systems and methodologies and be able to apply them. A list of the learning objectives and the verbs that appear in the syllabus learning outcomes and examinations, follows: Learning objectives 1

Knowledge What you are expected to know

Verbs used

Definition

List State

Make a list of Express, fully or clearly, the details of/ facts of Give the exact meaning of

Define 2

Comprehension What you are expected to understand

Describe Distinguish Explain Identify Illustrate

Communicate the key features of Highlight the differences between Make clear or intelligible/State the meaning of Recognise, establish or select after consideration Use an example to describe or explain something

x Syllabus Guidance, Learning Objectives and Verbs 3

Application How you are expected to apply your knowledge

Apply Calculate/ compute Demonstrate Prepare Reconcile Solve Tabulate

4

5

Analysis How you are expected to analyse the detail of what you have learned

Evaluation How you are expected to use your learning to evaluate, make decisions or recommendations

Analyse Categorise Compare and contrast Construct Discuss Interpret

To put to practical use To ascertain or reckon mathematically To prove with certainty or to exhibit by practical means To make or get ready for use To make or prove consistent/ compatible Find an answer to Arrange in a table

Produce

Examine in detail the structure of Place into a defined class or division Show the similarities and/or differences between To build up or compile To examine in detail by argument To translate into intelligible or familiar terms To create or bring into existence

Advise Evaluate Recommend

To counsel, inform or notify To appraise or assess the value of To advise on a course of action

Examination Techniques

Computer-based examinations 10 Golden rules 1 Make sure you are familiar with software before you start the exam. You cannot speak to invigilator once you have started. 2 These exam practice kits give you plenty of exam style questions to practice. 3 Attempt all questions, there is no negative marking. 4 Double check your answer before you put in final alternative. 5 On multiple choice questions, there is only one correct answer. 6 Not all questions will be MCQs – you may have to fill in missing words or figures. 7 Identify the easy questions first, get some points on the board to build up your confidence. 8 Try and allow five minutes at the end to check your answers and make any corrections. 9 If you don’t know the answer, try process of elimination. Sadly there is no phone a friend!! 10 Take scrap paper, pen and calculator with you. Work out answer on paper first if it is easier for you.

Computer-based assessment CIMA has introduced computer-based assessment (CBA) for all subjects at Certificate level. The website says Objective questions are used. The most common type is ‘multiple choice’, where you have to choose the correct answer from a list of possible answers, but there are a variety of other objective question types that can be used within the system. These include true/false questions, matching pairs of text and graphic, sequencing and ranking, labelling diagrams and single and multiple numeric entry. Candidates answer the questions by either pointing and clicking the mouse, moving objects around the screen, typing numbers, or a combination of these responses. Try the online demo at http://www.cimaglobal.com/to see how the technology works. The CBA system can ensure that a wide range of the syllabus is assessed, as a pre-determined number of questions from each syllabus area (dependent upon the syllabus weighting for that particular area) are selected in each assessment. xi

xii Examination Techniques In every chapter of this study system, we have introduced these types of questions but obviously we have to label answers A, B, C and so on rather than using click boxes. For convenience, we have retained quite a lot of questions where an initial scenario leads to a number of sub-questions. There will be questions of this type in the CBA but they will rarely have more than three sub-questions. In all such cases the answer to one part does not hinge upon a prior answer. There are two types of questions which were previously involved in objective testing in paper-based exams and which are not at present possible in a CBA. The actual drawing of graphs and charts is not yet possible. Equally there will be no questions calling for comments to be written by students. Charts and interpretations remain on many syllabi and will be examined at Certificate level but using other methods. For further CBA practice, CIMA Publishing has produced CIMA Inter@ctive CD-ROMs for all Certificate level subjects. These products use the same software as found in the real computer-based assessment and are available at www.cimapublishing.com.

Business mathematics and computer-based assessment The assessment for Business Mathematics is a one and a half hours computer-based assessment (CBA) comprising approximately 35 questions, with one or more parts. Single part questions are generally worth 2 marks each, but two and three part questions may be worth 4 or 6 marks. There will be no choice and all questions should be attempted if time permits. CIMA are continuously developing the question styles within the CBA system and you are advised to try the online website demo, to both gain familiarity with assessment software and examine the latest style of questions being used. A small amount of tolerance has been built into marking numerical answers in the assessments but it is important to pay great attention to any instructions given about rounding. You will find a brief discussion on rounding in Chapter 1. An instruction to round does not mean that you should work throughout with rounded figures. On the contrary, it may be expected that you will work with sufficient accuracy to give a final rounded answer as instructed. Accuracy is also essential in general since it is no longer possible to obtain marks for a correct method with a minor arithmetical error. In a CBA, you will be expected to understand and be able to write formulae in computer notation, using * to indicate multiplication and ^ to introduce a power. See Chapter 1 for an introduction to this notation. The mathematical tables and formulae given in this CIMA Study System will be available on paper when you sit for the CBA. Logarithm tables are also available.

The Business Mathematics syllabus Syllabus overview Syllabus subject C5, Business Mathematics, is a foundation study in mathematical and statistical concepts and techniques. The first two sections, Basic Mathematics and Summarising and Analysing Data, include techniques which are fundamental to the work of the management accountant. The third section covers basic probability and is needed because management accountants need to be aware of and be able to estimate the risk and uncertainty involved in

Examination Techniques xiii the decisions they make. The fourth section is an introduction to financial mathematics, a topic that is important to the study of financial management. Finally, there is an introduction to the mathematical techniques needed for forecasting, necessary in the area of business planning. Aims This syllabus aims to test the candidate’s ability to: • • • • • •

explain and demonstrate the use of basic mathematics including formulae and ratios; identify reasonableness in the calculation of answers; identify and apply techniques for summarising and analysing data; explain and demonstrate the use of probability where risk and uncertainty exist; explain and apply financial mathematical techniques; explain and demonstrate techniques used for forecasting.

Computer-based assessment The assessment is 90 minutes and comprises 35 compulsory questions with one or more parts. A varied range of objective test questions are used.

Learning outcomes and syllabus content (i)

Basic mathematics Study weighting: 10% learning outcomes On completion of their studies students should be able to: • • • • • •

demonstrate the order of operations in formulae, including the use of brackets, negative numbers, powers and roots; calculate percentages and proportions; calculate answers to appropriate significant figures or decimal places; calculate maximum absolute and relative errors; solve simple equations, including two-variable simultaneous equations and quadratic equations; prepare graphs of linear and quadratic equations.

Syllabus content • • • • (ii)

use of formulae; percentages and ratios; rounding of numbers; basic algebraic techniques and the solutions of equations – including simultaneous and quadratic equations.

Summarising and analysing data Study weighting: 25% learning outcomes On completion of their studies students should be able to: • •

explain the difference between data and information; explain the characteristics of good information;

xiv

Examination Techniques • • • • • • • •

• • •

explain the difference between primary and secondary data; identify the sources of secondary data; explain the different methods of sampling and identify where each is appropriate; tabulate data and explain the results; prepare a frequency distribution from raw data; prepare and explain the following graphs and diagrams: bar charts, time series graphs (not Z charts), histograms and ogives; calculate and explain the following summary statistics for ungrouped data: arithmetic mean, median, mode, range, standard deviation and variance; calculate and explain the following summary statistics for grouped data: arithmetic mean, median (graphical method only), mode (graphical method only), range, semi-interquartile range (graphical method only), standard deviation and variance; calculate and explain a simple index number, fixed-base and chain-base series of index numbers; use index numbers to deflate a series and explain the results; calculate a simple weighted index number. Candidates will not have to decide whether to use base or current weights.

Syllabus content • • • • • • • • (iii)

data and information; primary and secondary data; probability sampling: simple random sampling, stratified, quota, systematic, multi-stage and cluster; tabulation of data; frequency distributions; graphs and diagrams: bar charts, time series graphs (not Z charts), bivariate graphs, histograms and ogives; summary measures for both grouped and ungrouped data; index numbers.

Probability Study weighting: 20% learning outcomes On completion of their studies students should be able to: • • • • • • • •

calculate a simple probability; demonstrate the use of the addition and multiplication rules of probability; calculate a simple conditional probability; calculate and explain an expected value; demonstrate the use of expected values to make decisions; explain the limitations of expected values; demonstrate the use of normal distribution and tables; demonstrate the application of the normal distribution to calculate probabilities.

Syllabus content • • • •

the relationship between probability, proportion and percentage; the addition and multiplication rules; expected values; normal distribution.

Examination Techniques xv (iv)

Financial mathematics Study weighting: 20% learning outcomes On completion of their studies students should be able to: • • • • • • •

calculate future values of an investment using both simple and compound interest; calculate an annual percentage rate of interest, given a quarterly or monthly rate; calculate the present value of a future cash sum using both a formula and tables; calculate the present value of an annuity using both a formula and tables; calculate loan/mortgage repayments and the value of an outstanding loan/mortgage; calculate the present value of a perpetuity; calculate the future value of regular savings (sinking funds) or find the savings given the future value, if necessary, using the sum of a geometric progression; • calculate the NPV of a project and use this to decide whether a project should be undertaken, or to choose between mutually exclusive projects; • calculate and explain the use of the IRR of a project. Syllabus content • • • • • • (v)

simple and compound interest; discounting to find the present value; annuities and perpetuities; loans and mortgages; sinking funds and savings funds; simple investment appraisal.

Forecasting Study weighting: 25% learning outcomes On completion of their studies students should be able to: • calculate the correlation coefficient between two variables and explain the value; • calculate the rank correlation coefficient between two sets of data and explain the value; • explain the meaning of 100r2 (the coefficient of determination); • demonstrate the use of regression analysis between two variables to find the line of best fit, and explain its meaning; • calculate a forecast of the value of the dependent variable, given the value of the independent variable; • prepare a time series graph and identify trends and patterns; • identify the components of a time series model; • calculate the trend using a graph, moving averages or linear regression, and be able to forecast the trend; • calculate the seasonal variations for both additive and multiplicative models; • calculate a forecast of the actual value using either the additive or the multiplicative model; • explain the difference between the additive and multiplicative models, and when each is appropriate; • calculate the seasonally adjusted values in a time series; • explain the reliability of any forecasts made. Syllabus content • correlation; • simple linear regression; • time series analysis – graphical analysis;

xvi

Examination Techniques

• calculation of trend using graph, moving averages and linear regression; • seasonal variations – additive and multiplicative; • forecasting.

Mathematical tables Probability A ✜ B A or B. A B A and B (overlap). P(B|A) probability of B, given A. Rules of addition If A and B are mutually exclusive: P(A ✜ B) P(A) P(B) If A and B are not mutually exclusive: P(A ✜ B) P(A) P(B) P(A B) Rules of multiplication If A and B are independent: P(A B) P(A) * P(B) If A and B are not independent: P(A B) P(A) * P(B|A) E(X) expected value probability * payoff Quadratic equations If aX^2 bX c 0 is the general quadratic equation, then the two solutions (roots) are given by: X

b √b2 4ac 2a

Discriptive statistics Arithmetic mean x

x fx ,x (frequency distribution) n f

Standard deviation SD

√

(x x)2 , SD n

√

fx2 x 2 (frequency distribution) f

Index numbers Price relative 100 P1/P0, Quantity relative 100 Q1/Q0 Price: W * P1/P0/ W * 100, where W denotes weights Quantity: W *; Q1/Q0/ W * 100, where W denotes weights

Time series Additive model Series Trend Seasonal Random Multiplicative model Series Trend * Seasonal * Random

Examination Techniques xvii

Linear regression and correlation The linear regression equation of y on x is given by: Y a bX or Y Y b (X X ) where b

nXY (X)(Y) Covariance (XY) Variance (X) nX2 (X)2

and a Y bX or solve

Y na bX XY aX bX2 Coefficient of correlation r

Covariance (XY)

√Var (X) Var (Y)

R (rank) 1

nXY (X)(Y)

√

(nX2

(X)2)(nY2 (Y)2)

6d2 n (n2 1)

Financial mathematics Compound interest (Values and Sums) Future value of S, of a sum X, invested for n periods, compounded at r% interest S X[1r]n Annuity Present value of an annuity of £1 per annum receivable or payable for n years, commencing in one year, discounted at r% per annum.

1 1 PV r 1 [1 r]n

Perpetuity Present value of £1 per annum, payable or receivable in perpetuity, commencing in one year, discounted at r% per annum. 1 PV r Note: Logarithm tables are also available when you sit for your assessment.

Mathematical Tables AREA UNDER THE NORMAL CURVE This table gives the area under the normal curve between the mean and a point Z standard deviations above the mean. The corresponding area for deviations below the mean can be found by symmetry.

0

Z

(x )

z

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.1 0.2 0.3 0.4

.0000 .0398 .0793 .1179 .1554

.0040 .0438 .0832 .1217 .1591

.0080 .0478 .0871 .1255 .1628

.0120 .0517 .0910 .1293 .1664

.0159 .0557 .0948 .1331 .1700

.0199 .0596 .0987 .1368 .1736

.0239 .0636 .1026 .1406 .1772

.0279 .0675 .1064 .1443 .1808

.0319 .0714 .1103 .1480 .1844

.0359 .0753 .1141 .1517 .1879

0.5 0.6 0.7 0.8 0.9

.1915 .2257 .2580 .2881 .3159

.1950 .2291 .2611 .2910 .3186

.1985 .2324 .2642 .2939 .3212

.2019 .2357 .2673 .2967 .3238

.2054 .2389 .2704 .2995 .3264

.2088 .2422 .2734 .3023 .3289

.2123 .2454 .2764 .3051 .3315

.2157 .2486 .2794 .3078 .3340

.2190 .2518 .2823 .3106 .3365

.2224 .2549 .2852 .3133 .3389

1.0 1.1 1.2 1.3 1.4

.3413 .3643 .3849 .4032 .4192

.3438 .3665 .3869 .4049 .4207

.3461 .3686 .3888 .4066 .4222

.3485 .3708 .3907 .4082 .4236

.3508 .3729 .3925 .4099 .4251

.3531 .3749 .3944 .4115 .4265

.3554 .3770 .3962 .4131 .4279

.3577 .3790 .3980 .4147 .4292

.3599 .3810 .3997 .4162 .4306

.3621 .3830 .4015 .4177 .4319

1.5 1.6 1.7 1.8 1.9

.4332 .4452 .4554 .4641 .4713

.4345 .4463 .4564 .4649 .4719

.4357 .4474 .4573 .4656 .4726

.4370 .4485 .4582 .4664 .4732

.4382 .4495 .4591 .4671 .4738

.4394 .4505 .4599 .4678 .4744

.4406 .4515 .4608 .4686 .4750

.4418 .4525 .4616 .4693 .4756

.4430 .4535 .4625 .4699 .4762

.4441 .4545 .4633 .4706 .4767

2.0 2.1 2.2 2.3 2.4

.4772 .4821 .4861 .4893 .4918

.4778 .4826 .4865 .4896 .4920

.4783 .4830 .4868 .4898 .4922

.4788 .4834 .4871 .4901 .4925

.4793 .4838 .4875 .4904 .4927

.4798 .4842 .4878 .4906 .4929

.4803 .4846 .4881 .4909 .4931

.4808 .4850 .4884 .4911 .4932

.4812 .4854 .4887 .4913 .4934

.4817 .4857 .4890 .4916 .4936

2.5 2.6 2.7 2.8 2.9

.4938 .4953 .4965 .4974 .4981

.4940 .4955 .4966 .4975 .4982

.4941 .4956 .4967 .4976 .4983

.4943 .4957 .4968 .4977 .4983

.4945 .4959 .4969 .4977 .4984

.4946 .4960 .4970 .4978 .4984

.4948 .4961 .4971 .4979 .4985

.4949 .4962 .4972 .4980 .4985

.4951 .4963 .4973 .4980 .4986

.4952 .4964 .4974 .4981 .4986

3.0 3.1 3.2 3.3 3.4 3.5

.49865 .49903 .49931 .49952 .49966 .49977

.4987 .4991 .4993 .4995 .4997

.4987 .4991 .4994 .4995 .4997

.4988 .4991 .4994 .4996 .4997

.4988 .4992 .4994 .4996 .4997

.4989 .4992 .4994 .4996 .4997

.4989 .4992 .4994 .4996 .4997

.4989 .4992 .4995 .4996 .4997

.4990 .4993 .4995 .4996 .4997

.4990 .4993 .4995 .4997 .4998

xviii

PRESENT VALUE TABLE Present value of £1 is (1 r)n where r interest rate; n number of periods until payment or receipt.

Interest rates (r)

Periods (n)

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

16%

17%

18%

19%

20%

1 2 3 4 5

.990 .980 .971 .961 .951

.980 .961 .942 .924 .906

.971 .943 .915 .888 .863

.962 .925 .889 .855 .822

.952 .907 .864 .823 .784

.943 .890 .840 .792 .747

.935 .873 .816 .763 .713

.926 .857 .794 .735 .681

.917 .842 .772 .708 .650

.909 .826 .751 .683 .621

.901 .812 .731 .659 .593

.893 .797 .712 .636 .567

.885 .783 .693 .613 .543

.877 .769 .675 .592 .519

.870 .756 .658 .572 .497

.862 .743 .641 .552 .476

.855 .731 .624 .534 .456

.847 .718 .609 .516 .437

.840 .706 .593 .499 .419

.833 .694 .579 .482 .402

6 7 8 9 10

.942 .933 .923 .914 .905

.888 .871 .853 .837 .820

.837 .813 .789 .766 .744

.790 .760 .731 .703 .676

.746 .711 .677 .645 .614

.705 .665 .627 .592 .558

.666 .623 .582 .544 .508

.630 .583 .540 .500 .463

.596 .547 .502 .460 .422

.564 .513 .467 .424 .386

.535 .482 .434 .391 .352

.507 .452 .404 .361 .322

.480 .425 .376 .333 .295

.456 .400 .351 .308 .270

.432 .376 .327 .284 .247

.410 .354 .305 .263 .227

.390 .333 .285 .243 .208

.370 .314 .266 .225 .191

.352 .296 .249 .209 .176

.335 .279 .233 .194 .162

11 12 13 14 15

.896 .887 .879 .870 .861

.804 .788 .773 .758 .743

.722 .701 .681 .661 .642

.650 .625 .601 .577 .555

.585 .557 .530 .505 .481

.527 .497 .469 .442 .417

.475 .444 .415 .388 .362

.429 .397 .368 .340 .315

.388 .356 .326 .299 .275

.350 .319 .290 .263 .239

.317 .286 .258 .232 .209

.287 .257 .229 .205 .183

.261 .231 .204 .181 .160

.237 .208 .182 .160 .140

.215 .187 .163 .141 .123

.195 .168 .145 .125 .108

.178 .152 .130 .111 .095

.162 .137 .116 .099 .084

.148 .124 .104 .088 .074

.135 .112 .093 .078 .065

16 17 18 19 20

.853 .844 .836 .828 .820

.728 .714 .700 .686 .673

.623 .605 .587 .570 .554

.534 .513 .494 .475 .456

.458 .436 .416 .396 .377

.394 .371 .350 .331 .312

.339 .317 .296 .277 .258

.292 .270 .250 .232 .215

.252 .231 .212 .194 .178

.218 .198 .180 .164 .149

.188 .170 .153 .138 .124

.163 .146 .130 .116 .104

.141 .125 .111 .098 .087

.123 .108 .095 .083 .073

.107 .093 .081 .070 .061

.093 .080 .069 .060 .051

.081 .069 .059 .051 .043

.071 .060 .051 .043 .037

.062 .052 .044 .037 .031

.054 .045 .038 .031 .026

Mathematical Tables xix

1%

xx Mathematical Tables

CUMULATIVE PRESENT VALUE OF £1 This table shows the present value of £1 per annum, receivable or payable at the end of each year for n years

Interest rates (r)

Periods (n)

1 (1 r)n . r

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

16%

17%

18%

19%

20%

1 2 3 4 5

.990 1.970 2.941 3.902 4.853

.980 1.942 2.884 3.808 4.713

.971 1.913 2.829 3.717 4.580

.962 1.886 2.775 3.630 4.452

.952 1.859 2.723 3.546 4.329

.943 1.833 2.673 3.465 4.212

.935 1.808 2.624 3.387 4.100

.926 1.783 2.577 3.312 3.993

.917 1.759 2.531 3.240 3.890

.909 1.736 2.487 3.170 3.791

.901 1.713 2.444 3.102 3.696

.893 1.690 2.402 3.037 3.605

.885 1.668 2.361 2.974 3.517

.877 1.647 2.322 2.914 3.433

.870 1.626 2.283 2.855 3.352

.862 1.605 2.246 2.798 3.274

.855 1.585 2.210 2.743 3.199

.847 1.566 2.174 2.690 3.127

.840 1.547 2.140 2.639 3.058

.833 1.528 2.106 2.589 2.991

6 7 8 9 10

5.795 6.728 7.652 8.566 9.471

5.601 6.472 7.325 8.162 8.983

5.417 6.230 7.020 7.786 8.530

5.242 6.002 6.733 7.435 8.111

5.076 5.786 6.463 7.108 7.722

4.917 5.582 6.210 6.802 7.360

4.767 5.389 5.971 6.515 7.024

4.623 5.206 5.747 6.247 6.710

4.486 5.033 5.535 5.995 6.418

4.355 4.868 5.335 5.759 6.145

4.231 4.712 5.146 5.537 5.889

4.111 4.564 4.968 5.328 5.650

3.998 4.423 4.799 5.132 5.426

3.889 4.288 4.639 4.946 5.216

3.784 4.160 4.487 4.772 5.019

3.685 4.039 4.344 4.607 4.833

3.589 3.922 4.207 4.451 4.659

3.498 3.812 4.078 4.303 4.494

3.410 3.706 3.954 4.163 4.339

3.326 3.605 3.837 4.031 4.192

11 12 13 14 15

10.368 11.255 12.134 13.004 13.865

9.787 10.575 11.348 12.106 12.849

9.253 9.954 10.635 11.296 11.938

8.760 9.385 9.986 10.563 11.118

8.306 8.863 9.394 9.899 10.380

7.887 8.384 8.853 9.295 9.712

7.499 7.943 8.358 8.745 9.108

7.139 7.536 7.904 8.244 8.559

6.805 7.161 7.487 7.786 8.061

6.495 6.814 7.103 7.367 7.606

6.207 6.492 6.750 6.982 7.191

5.938 6.194 6.424 6.628 6.811

5.687 5.918 6.122 6.302 6.462

5.453 5.660 5.842 6.002 6.142

5.234 5.421 5.583 5.724 5.847

5.029 5.197 5.342 5.468 5.575

4.836 4.988 5.118 5.229 5.324

4.656 4.793 4.910 5.008 5.092

4.486 4.611 4.715 4.802 4.876

4.327 4.439 4.533 4.611 4.675

16 17 18 19 20

14.718 15.562 16.398 17.226 18.046

13.578 14.292 14.992 15.679 16.351

12.561 13.166 13.754 14.324 14.878

11.652 12.166 12.659 13.134 13.590

10.838 11.274 11.690 12.085 12.462

10.106 10.477 10.828 11.158 11.470

9.447 9.763 10.059 10.336 10.594

8.851 9.122 9.372 9.604 9.818

8.313 8.544 8.756 8.950 9.129

7.824 8.022 8.201 8.365 8.514

7.379 7.549 7.702 7.839 7.963

6.974 7.120 7.250 7.366 7.469

6.604 6.729 6.840 6.938 7.025

6.265 6.373 6.467 6.550 6.623

5.954 6.047 6.128 6.198 6.259

5.668 5.749 5.818 5.877 5.929

5.405 5.475 5.534 5.584 5.628

5.162 5.222 5.273 5.316 5.353

4.938 4.990 5.033 5.070 5.101

4.730 4.775 4.812 4.843 4.870

1

Formulae

Concepts, definitions and short-form questions 1

Simplify the following expressions (i) (ii) (iii) (iv) (v)

2

Expand the following expressions (i) (ii) (iii) (iv) (v)

3

2a2 a3 3a2 2a3 3x3 12x4 (3a2b)3 (2a2x)2 when a 2 and x 3


5

5(2x 3y 4z) x(2 3x) 4(x 2y 3z) 3x(2x 4y 3z) 4y(2y 4y 3z)


4

5a 6b 2a 3b 4x 3x 2x x 3b 4c 2b 5c 2a 3x 4y 2a 2y 3x 3x 9y 4x 5y 2z

x5 x2 x9 x7 15x5 3x4 a6 a6 a5 a7

Calculate the following values (i) (ii) (iii) (iv) (v)

30.5 20.5 41.5 103 360.5 1

2 Exam Practice Kit: Business Mathematics 6 Calculate (i) (ii) (iii) (iv) (v) 7

2 3 4 5 3 7 8 16 1 1 3 5 3 2 4 5 15⁄7 32⁄3

(i) How many parts does a log consist of? (ii) What is the mantissa? (iii) What is the purpose of the antilog table?

√

If C 10, P 6, R 0.2, D 600

√

If C £20, D 24,000, H £6

2DC PR What is the value of Q?

8 In the formula Q

2CD H What is the value of Q?

9 In the formula Q

10 State whether the following statements are true or false. (i) It does not matter in which order multiplications are carried out if there are brackets shown. (ii) The top of a fraction is called the denominator. (iii) The bottom of a fraction is called the numerator. (iv) When two or more powers of the same number are multiplied, the individual indices must be added. (v) The decimal fraction of a log number is called the mantissa. (vi) The whole number of a log is called the characteristic. (vii) The cube root of 64 is 4. (viii) A negative index is calculated by taking the inverse of the number. (ix) 15/8 is a larger number than 1.75. (x3)3 (x) The numerical value of when x 5 is 25 x7

Concepts, definitions and short-form solutions 1

(i) (ii) (iii) (iv) (v)

(5 2)a (6 3)b 7a 3b (4 3 2 1)x 4x 3b 4c 2b 5c b 9c (2 2)a (3 3)x (4 2)y 4a 2y (3 4)x (9 5)y 2z x 14y 2z

2

(i) (ii) (iii) (iv) (v)

10x 15y 20z 2x 3x2 4x 8y 12z 6x2 12xy 9xz 8y2 16y2 12yz

Formulae 3 3

(i) (ii) (iii) (iv) (v)

2 a(23) 2a5 3 2 a(23) 6a5 3 12 x(34) 36x7 33 (a2)3 x b3 27a6b3 (2a2x)2 22 24 32 4 16 9 576

4

(i) (ii) (iii) (iv) (v)

x(52) x3 x(97) x2 5x(54) 5x a(66) a0 1 a(57) a2 1/a2

5

(i) (ii)

√3 1.732 √2 1.414 3 √4 √64 8

(iii) (iv) (v) 6

(i) (ii) (iii) (iv) (v)

7

8

9

10

(i) (ii) (iii)

1 1 0.001 3 10 1,000 1 1 √36 6 15 8 23 3 1 20 20 20 20 14 3 11 16 16 16 1 1 1 3 5 15 3 5 15 17/8 4 2 8 115⁄21 314⁄21 58⁄21 2 the decimal fraction to convert back from the log to the number it represents

√2 600 10

6 0.2 so Q 100

100

√2 £20 24,000 £6 so Q 400 (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

false false false true true true true true true true

400

4 Exam Practice Kit: Business Mathematics

Multiple choice questions 1

6a 6b 2a 3b is equal to A B C D

2

3a3 4a4 is equal to A B C D

3

6a 3b 12ab 6ab 12a 18b 8a 3b

12a7 12a12 7a7 7a12

The number 82 is equal to A 16 B 6 C √8 1 D 64

4

The number 360.5 is equal to A 18 B 9 C 6 1 D 6

5

In the statement Z X Y which of the following statements is incorrect? A B C D

6

The antilog of 261 3.964 is A B C D

7

Z is greater than X but less than Y Y is greater than both X and Z X is greater than Z Z is lower than both X and Y

0261 03964 24166 30147

The statement X Y implies that A B C D

X is less than Y X is less than or equal to Y X is greater than Y X is greater than or equal to Y

Formulae 5 8 If a 2 and b 3, x 7 and y 8 b a then is equal to x y A B C D

37 56 2 9 14 28 5 15

9 The numeric value of the expression (x3)3 when x 5 is x A 0 B 5 C 25 D 125 10 Which of the following operations will not affect the order in which the numbers appear? A B C D

addition and multiplication addition and subtraction subtraction and division division and multiplication

Multiple choice solutions 1

6a 6b 2a 3b is equal to 8a 3b. so D

2

We need to multiply the numbers and add the powers so 3a3 4a4 12a7. so A

3

82

1 1 2 8 64

so D 4 360.5

1 1 √36 6

so D 5

Statement A is incorrect since Z is less than both X and Y. so A

6 log 261 2.4166 log 3.964 0.5981 add 3.0147 so D

6 Exam Practice Kit: Business Mathematics 7 If X Y, this implies that X is less than or equal to Y. so B 8

2 3 16 21 37 7 8 56 56 56 so A Note: 2/7 3/8

multiplied by 8 to get LCD multiplied by 7 to get LCD

9 A similar question to Q2, top equation becomes x9, so

x9 x2 x7

if x 5 then x2 25 so C 10 Addition and multiplication do not affect the order in which the numbers appear. so A

Equations and Graphs

2


Solve the following equations (i) 3x 4y 25 (ii) x y 10 (iii) 2x 3y 42 y x (iv) 7 3 2

2

4x 5y 32 x 4y 0 5x y 20 y 2x 7 3 6

Solving simultaneous equations with three unknowns 2x 3y 4z 9 (1) 3x 2y 3z 3 (2) 4x 5y 2z 25 (3)

3

Factorise the following expressions (i) (ii) (iii) (iv) (v)

4

am bm an bn 12a2m3 15am5 x2 4x 12 10p2 11pq 6q2 3x2 6x

Quadratic equations. Solve the following (i) 9x2 30x 25 0 (ii) 3x2 20x 15 40 (iii) x2 6x 9 25

Questions 5, 6, 7 and 8 are based on the following data The group economist believes that if their new product was priced at £100 they will sell 500 units per week. If price was £50, they would sell 800 units per week. The production department has overheads of £10,000 per week and variable cost per unit is £7.50 per unit. 5

Derive an expression for Total Cost.

6

Derive an expression for Total Revenue.

7

Derive an expression for Profit.

8

At what output is profit maximised? 7

8 Exam Practice Kit: Business Mathematics 9 Solve the following by factorisation if possible, otherwise use the formula. x2 5x 6 0 x2 6x 7 0 x2 6x 9 0 2x2 5x 20 0

(i) (ii) (iii) (iv)

10 State whether the following statements are true or false? In the equation y a bx, y is the dependant variable. Simultaneous equations with three unknown variables cannot be solved. The shape of a linear demand curve is a straight line. Quadratic equations can never be solved by factorisation. If b2 4ac is zero, there is only one solution to the equation. If b2 4ac is positive, there are no real solutions. If x y 9 then both numbers must be positive. If the length of the box is twice the size of the square end of a square-ended rectangular box, then the volume could be calculated using the formula 2x3. b √(b2 4ac) (ix) The formula used to solve quadratic equations is x . 2a (x) Quadratic equations cannot be solved by using a graph.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)


(i) 3x 4y 25 multiply by 4 (1) 4x 5y 32 multiply by 3 (2) 12x 16y 100 12x 15y 96 subtracting (1) from (2) becomes y 4 3x 16 25 x3 so x 3, y 4 (ii)

x y 10 (1) x 4y 0 (2) subtracting (2) from (1) becomes 5y 10 i.e. y (4y) 5y If y 2 then x 8

(iii)

2x 3y 42 (1) 5x y 20 (2) multiply equation (2) by 3 15x 3y 60 2x 3y 42 add 17x 102 x6 so x 6, y 10

(iv)

x y 2x y 7 7 3 2 3 6 multiply everything by 6 2x 3y 42 (1) 4x y 42 (2)

Equations and Graphs 9 multiply equation (2) by 3 2x 3y 42 12x 3y 126 14x 168 x 12 If x 12, then y 6 2

2x 3y 4z 9 (1) 3x 2y 3z 3 (2) 4x 5y 2z 25 (3) Step 1 eliminate x 6x 9y 12z 27 equation (1) 3 6x 4y 6z 6 equation (2) 2 subtract 13y 18z 21 (4) 4x 6y 8z 18 equation (1) 2 4x 5y 2z 25 (3) subtract y 10z 7 (5) multiply equation (5) by 13 13y 130z 91 (6) 13y 18z 21 112z 112 z 1 y 10z 7 y 10 7 y 7 10 y3 2x 3y 4z 9 2x 9 4 9 2x 4 x2 solution x 2, y 3, z 1

3

(i) (ii) (iii) (iv) (v)

m(a b) n(a b) 3am3(4a 5m2) (x 6)(x 2) (5p 2q)(2p 3q) 3x(x 2)

√

(30)2 4 9 25 0 29 5 30 √0 18 3 (20)2 4 3 15 (ii) (20) 0 29 20 14.83 20 √220 20.82 or 19.18 18 18 √62 4 1 9 0 (iii) 6 21 √0 3 6 2 5 Total cost £10,000 7.5q 4

(i) (30)

√

10 Exam Practice Kit: Business Mathematics 6 Let price per unit be £p/unit when x 500 p 100 500 a bp when x 800 p 50 800 a bp 500 a 100b 800 a 50b 300 50b b 6 500 a 100 (6) 500 a 600 a 1,100 x 1,100 6p 6p 1,100 x 1,100 x x p 183.33 6 6 Sales revenue p x x so R x 183.33 6 2 x R 183.33x 6

7 Profit R T

183x

x2 6

(10,000 7.5x)

8 Profit is maximised where marginal cost marginal revenue. 9

10

(i) Factorise x 2 or 3 (ii) Does not factorise x 1.59 or 4.41 (iii) Factorises x 3 twice (iv) Does not factorise x 2.15 or 4.65 (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

true false true false true false false true true false


A square-ended rectangular box has a volume of 1,458 cm3. The length of the box is twice that of one side of the square end. One side of the square end measures A B C D

6 cm 9 cm 18 cm 24 cm

Equations and Graphs 11

Questions 2, 3 and 4 are based on the following information The marketing department estimates that if the selling price of the new product is set at £40 per unit, sales will be 400 units per week. If the selling price is £20 per unit, sales will be 800 units per week. The production department estimates that variable costs will be £7.50 per unit and fixed costs £10,000 per week. 2

The cost equation is A B C D

3

£10,000 £7.5x £10,000 £7.5x £10,000 £40 £10,000 £75x

The sales revenue equation is A 400 9 B 1,200 9 x2 C 60x 20 D 60x x2

4

The profit equation is x2 52.5x 10,000 20 B x 52.5 10,000 C x2 52.5x 10,000 D x2 52.5x 10,000 A

5

If 3x 4y 25 and 10x 2y 38 what are the values of x and y? A B C D

6

x3 x3 x4 x5

y3 y4 y5 y4

If 9x2 30x 25 0 then x is equal to 2 3 B 1 5 C 3 D 2 A

7

The shape of a graph of linear equation will be A B C D

8

U shape straight line L shape depends on the linear equation

If b2 4ac is positive then A B C D

there is only one solution there are two possible solutions there are no real solutions impossible to determine without knowing their values

12 Exam Practice Kit: Business Mathematics 9

2x 3y 4z 9 3x 2y 3z 3 4x 5y 2z 25 the values of x, y and z are

In the equations

A B C D

x 2 y 3 z 1 x1 y3 z2 x2 y1 z3 x3 y2 z1

10 If 6x2 12x 4(5x 2) then the values of x are A B C D

2 or 2 3 3 or 1 2 3 or 1 2 2 or 2 3


Volume of rectangular box is equal to length height depth. 1,458 2x x x 1,458 2x 3 729 x 3 3 √729 x x9 so B

2

Total cost Fixed cost Variable cost £10,000 £7.5x so A

3

x a bp since graph of this function is linear when x 400 p 40 400 a 40b (1) when x 800 p 20 800 a 20b (2) subtracting (1) from (2) we have 400 20b 400 b 20 so b 20 using equation (1) 400 a 40 (20) 400 a 800 400 800 a a 1,200 so x 1,200 20p 20p 1,200 x

Equations and Graphs 13 1,200 x x 60 20 20 Sales revenue x p x so x 60 20 x2 R 60x 20 so C p

4

The profit equation is total revenue total cost x2 (10,000 7.5x) 60x 20 x2 52.5x 10,000 20 so A

5

3x 4y 25 (1) 10x 2y 38 (2) Multiply equation (2) by 2 20x 4y 76 3x 4y 25 17x 51 x3 3x 4y 25 9 4y 25 4y 16 y4 so x 3 and y 4 so B

6

Using the formula b √b2 4ac x 2a where a 9 b 30 c 25 (30) √900 900 x 18 30 0 18 5 3 so C

7

The shape of a graph of a linear equation will be a straight line. so B

8

If b2 4ac is zero, there is only one solution If b2 4ac is positive, there are two solutions If b2 4ac is negative, there are no solutions so B

9

Eliminate x (either of the other variables would do) between (1) and (2): 3 (1): 6x 9y 12z 27 2 (2): 6x 4y 6z 6

14 Exam Practice Kit: Business Mathematics Subtract: 13y 18z 21 (4) Eliminate x between (3) and (1), ((2) could have been used instead of (1)): 2 (1): 4x 6y 8z 18 (3): 4x 5y 2z 25 Subtract: y 10z 7 (5) Multiply equation (5) by 13 13y 130z 91 (6) Subtract (4) from (6) 13y 130z 91 (6) 13y 18z 21 (4) 112z 112 Rearrange the equation to obtain a value for z. 112z 112 112 z 112 z 1 Substitute the value of z into equation (5) y 10z 7 y 10(1) 7 y 10 7 y 7 10 y3 Due to there being three unknowns, substitute the values for y and z into equation (1) to obtain a value for x. 2x 3y 4z 9 (1) 2x 3 (3) 4 (1) 9 2x 9 4 9 2x 9 9 4 2x 4 4 x 2 x2 Check in either (2) or (3): from(2): 3 2 2 3 3 (1) 3 6633 33 Hence the solution is x 2, y 3, z 1 so A 10 6x2 12x 4(5x 2) 6x2 12x 20x 8 6x2 8x 8 0 (8) √(8)2 (4 6 (8)) x 26 8 √64 192 12 16 8 12 12 2 x or x 2 3 so A

Percentages, Ratios and Proportions

3


If VAT is levied on goods and services at 17 1/2%, how much VAT is paid on goods costing VAT inclusive? (i) (ii) (iii) (iv) (v)

£117.50 £150 £200 £250 £500

2

Equipment is sold for £240 and the cost price is £200. Distinguish between the gross profit and the mark-up.

3

If sales are £500 and gross profit is £200, express the gross profit as (i) a fraction (ii) a decimal (iii) a percentage (iv) a ratio

4

James, Fred and Martin are in a business partnership and over the past year have made a profit of £50,000. They have agreed to split profit in the ratio of 5:7:8. How is the profit distributed between them?

5

Three balls (red, white and blue) are put into a bag. How many different ways are there of pulling the balls out?

6

Company Y offers its customers a 12% discount on all orders over £500 and 15% on all orders over £1,000. If customer A spends £1,200 and customer B £650, how much discount do they end up giving away.

7

State whether the following statements are true or false. (i) To convert a fraction into a % multiply by 100. (ii) A proportion cannot be measured as a decimal. (iii) If 8 dogs are picked for the final of a dog show, and 3 are to be picked 1st, 2nd and 3rd. There are 500 possible results. 15

16 Exam Practice Kit: Business Mathematics (iv) A person pays £228 for goods having received a discount of 5%. The undiscounted price was £250. (v) x% of 300 3x (vi) An article is sold for £300 VAT inclusive. The vendor receives £300 which is credited to sales. 8

What is a permutation?

9

What is a combination?


(i) (ii) (iii) (iv) (v)

2

Gross profit and mark-up are both £40 but margins are different. 40 1 GP 240 6 1 40 Mark up 200 5

3

4

£17.50 £22.34 £29.78 £37.23 £74.46

200 2 500 5 (ii) .4 (iii) 40% (iv) 4:10 (i)

5 £12,500 20 7 Fred £17,500 20 8 Martin £20,000 20 James

5

Red Red White White Blue Blue

White Blue Red Blue Red White

6

Customer A discount 15% of 1,200 180 Customer B discount 12% of 650 78 Total discount given

7

(i) (ii) (iii) (iv) (v) (vi)

true false false false true false

Blue White Blue Red White Red

£258

Percentages, Ratios and Proportions 17 8

A permutation is a number of selected items from a larger group of items, where the order in which the items are selected is significant. Question 5 is an example of a permutation.

9

A combination is a number of items selected from a larger group of items, where the order in which those items are selected is not significant.


Equipment is sold for £240 and makes a profit of 20% on cost. What is the profit price? A B C D

2

If sales are £500 per week and cost of sales are £300 per week, gross profit expressed as a percentage is A B C D

3

15% 19% 23% 28%

An article in a sales catalogue is priced at £298 including VAT at 17.5%. The ex-VAT price of the product is A B C D

6

£10,000 £12,000 £14,000 £16,000

If the population of Westend on Sea is 278,000 and 54,000 are of school age, what proportion of the population is of school age? A B C D

5

10% 20% 30% 40%

Alex, Dave and John are in partnership and profits are split in the ratio 7:6:5. If profit for the year is £36,000, how much does Alex receive? A B C D

4

£10 £20 £30 £40

£247.34 £253.62 £255.00 £280.50

x% of 200 equals x 200 B x1/2 C 200 x D 2x A

18 Exam Practice Kit: Business Mathematics 7 The number 10.37951 is equal to what if taken to two decimal places? A B C D

10.4 10.3 10.37 10.38

8 Three years ago Smith Bros. purchased a van for £12,000. If they depreciate the vehicle by 25% on a reducing balance basis, the value of the vehicle at the end of year 3 is A B C D

£6,000 £5,550.75 £5,062.50 £4,750.25

9 An audit team is made up of a manager, two seniors and four juniors. If there are 10 managers, 15 seniors 20 juniors how many different audit teams could be formed from these numbers? A B C D

20 505,325 5,087,250 impossible to determine

10 City and United play each other twice over the season. How many permutations are there in the scores? A B C D

2 6 9 12


If cost price 100% then selling price is 120% of cost 120% of cost £240 100 100% of cost 240 120 £200 Profit £240 £200 £40 so D

2 Gross profit

£200 40% £500

so D 3

Total profit to be distributed £36,000 7 Alex receives £36,000 £14,000 18 so C

Percentages, Ratios and Proportions 19 4

27 54,000 19% 278,000 139 so B

5 Ex VAT price

£298 100 £253.62 117.5

so B 6 Suppose x 10% 10% of 100 10 10% of 200 20 So x% of 200 is 2x of 100 so D 7 We can eliminate A and B since they only go to one decimal place. At three decimal places 0.379 is closer to 0.38 than 0.37. so D 8

Depreciation Value year 1 Value year 2 Value year 3 so C

9,000 6,750 5,062.50

3,000 2,250 1,687.50

9 No. of managers 10 15 14 105 No. of seniors 21 20 19 18 17 No. of assistants 4,845 4321 Therefore number of different audit teams 10 105 4,845 5,087,250 so C 10 Ist game City City City Draw Draw Draw United United United 9 so C

2nd game United City Draw United City Draw United City Draw


4


Complete the following phrase or sentence. (i) (ii) (iii) (iv) (v) (vi)

2

A ............. variable is one that can assume any value. A ............. variable is one that can only assume certain values. An ............. variable is one which is not affected by changes in another. A ............. variable is affected by changes in another. When individuals are rounded in the same direction, this is a ............. error. When individuals are rounded in either direction this is an ............. error.

If 4.782 is added to 3.42 (i) calculate the highest possible value (ii) calculate the lowest possible value

3

If 3.42 is subtracted from 4.782 (i) calculate the highest possible value (ii) calculate the lowest possible value

4

If 3,260 is multiplied by 125 (i) calculate the highest possible value (ii) calculate the lowest possible value

5

Find the highest and lowest value of 3,260 125.

6

Calculate 32.6 4.32 and the error if the figures have been rounded to three significant figures.

7


8


9

Calculate the result of 32.6 4.32 and its error limits.

10 A product was priced at £56.99 has been reduced to £52.49. To two decimal places, the percentage reduction in price is? 20



(i) (ii) (iii) (iv) (v) (vi)

continuous discrete independent dependent biased unbiased

2 Highest 4.7825 3.425 8.2075 Lowest 4.7815 3.415 8.1965 3 Highest 4.7825 3.425 1.3575 Lowest 4.7815 3.415 1.3665 4 Highest 3,265 125.5 409,757.5 Lowest 3,255 124.5 405,247.5 5 Highest 3,265 124.5 26.22490 Lowest 3,255 125.5 25.93626 6 (32.6 0.05) (4.32 0.05) 36.82 7 (32.6 0.05) (4.32 0.05) 28.28 8 Highest 32.65 4.325 141.211 Lowest 32.55 4.315 140.453 so 140.83 0.38 9 32.6 4.32 7.5463 Relative error 0.269% 0.269 7.5463 Absolute error 100 0.0203 so 7.55 0.02 10

56.99 52.49 100 7.90% 56.99


A biased error arises when A B C D

2

individual items are rounded in the same direction individual items are rounded in either direction individual items are rounded in the opposite direction individual items are not rounded

An unbiased error arises when A B C D

individual items are rounded in the same direction individual items are rounded in either direction individual items are rounded in the opposite direction individual items are not rounded

21


A discrete variable is one which A B C D

4

A relative error is A B C D

5

can assume any value can only assume certain specific values is not affected by changes in other variables is affected by changes in other variables

an absolute error an absolute error expressed as a percentage a numerical error none of the above

A product was priced at £117.58 and has been reduced to £105.26. To two decimal places the percentage reduction in price was A B C D

9.52% 9.93% 10.00% 10.48%


A biased error arises when individual items are rounded in the same direction. so A

2

An unbiased error arises when individual items are rounded in either direction. so B

3

A discrete or continuous variable is one that can assume any value. so A

4

A relative error is an absolute error expressed as a percentage. so B

5

£117.58 £105.26 100 10.48% £117.58 so D

Financial Mathematics

5


A boy is given £100 from his grandmother on the 1st January each year. On 31st December simple interest is credited at 10% which he withdraws to spend. How much will be in the account on 31st December after five years?

2

If the boy kept the interest and credited it to his account each year, how much would be in the account on 31st December after five years if £100 was invested in year 1 but no more payments received after that?

3

A new machine costs £5,000 and is depreciated by 8% per annum. What is the book value of the machine after five years?

4

A new machine costs £8,000 and lasts 10 years and has a scrap value of £100. What is the annual rate of compound depreciation?

5

Dougie is saving to pay for his daughter’s wedding in five years’ time. He puts £400 per year in the bank which will earn interest at 9%. The wedding is expected to cost £3,000. Will he have saved up enough by then?

6

How much needs to be invested now at 6% per annum to provide an annuity of £5,000 per annum for ten years commencing in five years’ time?

7

Calculate the annual repayment on a bank loan of £50,000 over eight years at 9% per annum.

8

How much needs to be invested now at 5% to yield an annual income of £10,000 in perpetuity?

9

An initial investment of £2,000 yields yearly cash flows of £500, £500, £600, £600 and £440 at the end of each year. At the end of year five, there is no scrap value. If capital is available at 12%, using discounted cash flow and internal rate of return assess whether the project should be accepted.

10

If a credit card company has an annual percentage rate (APR) of 30% how much interest are they charging a customer each month? 23



Year

Investment £

1 2 3 4 5 So £500 2

Year

100 200 300 400 500 Principal £

Interest £

Total £

100 110 121 133.10 146.41

10 11 12.10 13.31 14.64

110 121 133.10 146.41 161.05

1 2 3 4 5 3

x £5,000 8 0.08n 5 100 D x(1r)n £5,000 (10.08)5 £5,000 0.6591 £3,295 r 8%

4

D £100 x £8,000 n 10 so 100 8,000 (1 r)10 100 (1 r)10 0.0125 8,000 10

(1 r) √0.0125 0.6452 r 1 0.6452 0.3548 35.48% 5

Assume first instalment is paid immediately Sn

A(Rn 1) R1

where A annual savings R 1.09 Sn is the amount saved after 6 years

n6

£400 (1.096 1) 1.09 £3,009

6

See annuity table. Check values of 6% at year 14 and year 4 PV £5,000 (year 14 year 4) £5,000 (9.295 3.465) £29,150

7

Again go straight to cumulative present value and look up the value for 9% at eight years 5.535 £50,000 so £9,033.42 per annum 5.535

8

Present value of perpetuity is £10,000

1 £200,000. 0.05

Financial Mathematics 25 9 Net present value Year 0 1 2 3 4 5

Cash flow £

DCF-12%

Present value

(2,000) 500 500 600 600 440

1 0.893 0.797 0.712 0.636 0.567

(2,000) 447 399 427 382 249

Net present value £96 Internal rate of return 8% Year 0 1 2 3 4 5

Cash flow £

DCF-8%

Present value

(2,000) 500 500 600 600 440

1 0.926 0.857 0.794 0.735 0.681

(2,000) 463 429 476 441 300

Net present value £109 IRR is between 12% and 8% BA NANA NB 109 (12% 8%) 8% 109 (96) 109 8% 4% 2.5

So IRR A

10.13%

Reject NPV because at 12% it is negative. Reject IRR because it is below 12%. 10 (1 r)12 1.30 There are 12 months in a year 12 So 1r √1.3 2.21 So r 2.21%


A credit card company is charging an annual percentage rate of 25.3%. This is equivalent to a monthly rate of A B C D

1.8 1.9 2.0 2.2


Johnny receives £1,000 per annum starting today and receives five such payments. If the rate of interest is 8% what is the net present value of this income stream? A B C D

3

A new machine costs £5,000 and is depreciated by 8% per annum. The book value of the machine in five years time will be A B C D

4

£1,200 one year from now £1,400 two years from now £1,600 three years from now £1,800 four years from now

How much would need to be invested today at 6% per annum to provide an annuity of £5,000 per annum for ten years commencing in five years’ time? A B C D

8

£6,250 £6,973 £7,247 £7,915

If interest rates are 8%, which is worth most at present values? A B C D

7

£1,200 one year from now £1,400 two years from now £1,600 three years from now £1,800 four years from now

A land lord receives a rent of £1,000 to be received over ten successive years. The first payment is due now. If interest rates are 8% then the present value of this income is equal to A B C D

6

£5,000 £4,219 £3,295 £2,970

Which is worth most, at present values, assuming an annual rate of interest of 12%? A B C D

5

£4,000 £4,100 £4,282 £4,312

£5,000 £19,000 £29,150 £39,420

What is the annual repayment on a bank loan of £50,000 over eight years at 9%? A B C D

£8,975 £9,033 £9,214 £9,416

Financial Mathematics 27 9 How much needs to be invested now at 5% to yield an annual income of £4,000 in perpetuity? A B C D

£80,000 £90,000 £100,000 £120,000

10 The main advantage of using a discounted cash flow approach to investment appraisal over more traditional methods is A B C D

it is easier to calculate it is easier to understand it will show a higher profit figure it is a technique which recognises the time value of money


(1 r)12 1.25 There are 12 months in a year. 12 so 1 r √1.25 1.9 so B

2

£1,000 (1 cumulative factor for year 4 at 8%) £1,000 (1 3.312) £4,312 so D

3

x £5,000 8 0.08 100 D x (1 r)n £5,000 (1 0.08)5 £5,000 0.6591 £3,295 so C

r 8%

4

£1,200/1.12 £1,071 £1,400/(1.12)2 £1,116 £1,600/(1.12)3 £1,139 £1,800/(1.12)4 £1,144 so D

5

NPV £1,000 (1 6.247) £1,000 7.247 £7,247 so C

6

A £1,200/1.08 £1,111.20 2 B £1,400/(1.08) £1,199.80 C £1,600/(1.08)3 £1,273.40 D £1,800/(1.08)4 £1,323.00 so D

28 Exam Practice Kit: Business Mathematics 7 Check cumulative present value table 6% year 14 9.295 6% year 4 3.465 subtract 5.830 £5,000 5.830 £29,150 so C 8 Let x annual repayment Present value of 8 repayments of x at 9% £50,000 From tables 5.535 x £50,000 £50,000 x £9,033 5.535 so B 9 A real life example of this is a pension. In other words you are living off the interest and the capital remains. 1 £4,000 £80,000 0.05 so A 10 The main advantage of using a discounted cash flow approach to investment appraisal is that it is a technique which recognises the time value of money. so D

Data Collection

6


Distinguish between data and information.

2

Explain the following terms (i) (ii) (iii) (iv) (v) (vi)

3

random sampling systematic sampling stratified sampling multi-stage sampling cluster sampling quota sampling

What are the four main methods that primary data can be collected? (i) (ii) (iii) (iv)

4

Distinguish between primary and secondary data.

5

What are the four most important things to consider when constructing a frequency distribution? (i) (ii) (iii) (iv)


Data consists of numbers, letters, symbols, facts, events and transactions which have been recorded but not yet processed into a form which is suitable for making decisions. Information is data which has been processed in such a way that it has meaning to the person who is receiving it. 29


Sampling (i) A random sample is a sample taken in such a way that every member of the population has an equal chance of being selected. (ii) If the population is known and a sample size of a certain number is required then one in so many items is selected. This is systematic sampling. (iii) If the population being sampled contains several well defined groups (called strata) e.g. if 20% of the population are pensioners, this method of stratified sampling ensures that a representative cross section of the strata in the population is obtained. (iv) Multi-stage sampling would be used if you were carrying out a national survey e.g. an opinion poll prior to the election. The country would be divided into areas, then town or cities, then roads and streets, etc. (v) Cluster sampling is similar to multi-stage sampling except that every house in a particular area would be visited rather than just a random sample. (vi) With quota sampling the person conducting the interview would be given a list or quota of certain individuals, for example, professional females aged 20–35 25%.

3

Survey methods (i) postal questionnaire (ii) personal interview (iii) telephone interview (iv) observation

4

Primary and secondary data Primary data is data which has been specifically collected for a particular enquiry. Secondary data is data which has been collected for some other enquiry.

5

Frequency distributions The four most important things are (i) number of classes (ii) class intervals or widths (iii) open ended class intervals (iv) class limits


Which of the following statements is correct? A B C D

2

data data information data information meaning data meaning information information meaning data

A sample taken in such a way that every member of the population has an equal chance of being selected is known as a A B C D

random sample systematic sample stratified sample multi-stage sample

Data Collection 31 3

Stratified sampling is often preferred to systematic sampling because A this method can be applied with the minimum of difficulty B this method ensures that a representative cross section of the sample in the population is obtained C it is carried out on a national basis D there is no difference between stratified and systematic sampling

4

Which of the following methods can be used to collect primary data? (i) postal questionnaire (ii) personal interview (iii) telephone interview (iv) observation A B C D

5

i, ii i, ii, iv i, iii, iv i, ii, iii, iv

A management accountant is checking invoices for errors. Every invoice has a serial number the first being 3001. She has decided to sample 1 in 5 invoices. After selecting a random number from 1 to 5 she will then check invoices numbered 3005, 3010, 3020, etc. This sampling method is termed A B C D

random stratified clustered systematic


The statement which is correct is data meaning information. so C

2

A sample taken in such a way that every member of the population has an equal chance of being selected is known as a random sample. so A

3

Stratified sampling is often preferred to systematic sampling because this method ensures that a representative cross section of the sample in the population is obtained. so B

4

Postal questionnaires, personal interviews, telephone interviews and observation are all used to collect primary data. so D

5

This is an example of systematic sampling. so D

Presentation of Data

7


In constructing graphs and diagrams, state six principles which should be followed. (i) (ii) (iii) (iv) (v) (vi)

2

State three types of bar charts. (i) (ii) (iii)

3

A farmer has land extending to 100 acres which comprises 43% wheat, 20% barley, 16% grass, 12% oats and 9% fallow. If these figures were drawn in a pie chart what would be the angle of each?

4

United won the league last season using 20 players. There were 40 league games. Players who played

Frequency

35–40 games 30–34 games 25–29 games 20–24 games 15–19 games 10–14 games less than 10 games

1 3 4 5 1 2 4

Construct a frequency distribution showing the number of times players played. 32

Presentation of Data 33 5

Write down the values of the following. (i) (ii) (iii) (iv) (v)

6

lower quartiles upper quartiles decile percentile 2nd decile

There are 100 packets of biscuits in a box with the following weights and frequencies Weights

Frequency

100 and less than 110 110 and less than 120 120 and less than 130 130 and less than 140 140 and less than 150 150 and less than 160 160 and less than 170 170 and less than 180 180 and less than 190 190 and less than 200

1 2 5 11 21 20 17 11 6 6

Draw a cumulative frequency curve. 7

The following data was extracted from the annual report of XY plc. Annual sales (£millions) 2001 2002 UK EC USA Australia

31.5 33.2 40.3 26.1 131.1

35.0 47.4 78.9 18.2 179.5

Show this information (i) in a pie chart (ii) a component bar chart.

Questions 8–10 are based on the following data Student marks (%) Over 80 70–79 60–69 50–59 40–49 30–39 20–29

Number of students 2 8 15 30 25 10 10 100

34 Exam Practice Kit: Business Mathematics 8 From the information above construct a histogram. 9 From the information above construct a frequency polygon. 10 From the information above construct an ogive.


Principles to be followed constructing graphs and diagrams (i) (ii) (iii) (iv) (v) (vi)

2

3

give the diagram a name state where data is sourced units of measurement must be stated scale must be stated axes must be clearly labelled neatness is essential

(i) simple (ii) component (iii) multiple 43 360 155 100 20 barley 360 72 100 16 grass 360 58 100 12 oats 360 43 100 9 72 fallow 360 100 360

(i)

wheat

(ii) (iii) (iv) (v)

4

Frequency

No. of players used by United over 40 league games

5

(i) (ii) (iii) (iv) (v)

6 5 4 3 2 1 0

>10 10–14 15–19 20–24 25–29 Frequency

30–34

games

35–40

25% through the cumulative frequencies 75% through the cumulative frequencies 10% of the population 1% of the population 20% through the cumulative frequencies

Presentation of Data 35 6 Class interval Weight 100 and less than 110 110 and less than 120 120 and less than 130 130 and less than 140 140 and less than 150 150 and less than 160 160 and less than 170 170 and less than 180 180 and less than 190 190 and less than 200

Frequency No. of articles

Cumulative Frequency

1 2 5 11 21 20 17 11 6 6

1 3 8 19 40 60 77 80 94 100

Cumulative frequency curve 120 100 80 60 40 20 0 100 110 120 130 140 150 160 170 180 190 > > > > > > > > > > 110 120 130 140 150 160 170 180 190 200

Note: Cumulative frequency is always plotted at the upper mathematical class limit. 7

(i) Pie chart

UK EC USA Australia

Sales

2001 Angles

Sales

2002 Angles

31.5 33.2 40.3 26.1 131.1

86.5 91.2 110.7 71.6 360.0

35.0 47.4 78.9 18.2 179.5

70.2 95.1 158.2 36.5 360.0

Radii of circles

√131.1 and 11.4 and So if radius for 2001 3 cm

Radius for 2002

√179.5 13.4

13.4 3 3.5 cm 11.4

2001 UK EC USA Australia


UK EC USA Australia

(ii)

Component bar charts

200 150 100 50 0

Australia USA EC UK Sales 2001

Sales 2002

35 30 25 20 15 10 5 0

Number passed

ov er 8 70 0 –7 60 9 –6 50 9 –5 40 9 –4 30 9 –3 20 9 –2 9

Number passed

8

Student marks

35 30 25 20 15 10 5 0

Number passed

ov er 80 70 –7 60 9 –6 50 9 –5 9 40 –4 30 9 –3 20 9 –2 9

Number passed

9

Student marks

35 30 25 20 15 10 5 0

9 –5 9 40 –4 9 30 –3 9 20 –2 9

9

50

–6

–7

60

70

80

Number passed

er ov

Number passed

10

Student marks

Presentation of Data 37


Which of the following is not a feature of a frequency distribution? A B C D

2

In a histogram in which one class interval is one and a half times as wide as the remaining classes, the height to be plotted in relation to the frequency for that class A B C D

3

1% of the population 10% of the population the majority of the population 90% of the population

Cumulative frequencies are plotted against A B C D

7

25% through the cumulative frequencies 75% through the cumulative frequencies 1% of the population 10% of the population

The decile represents A B C D

6

£135,000 £137,500 £142,000 £145,000

An ogive represents a cumulative frequency distribution on which can be shown ranges of values containing given proportions of the total population. The upper quartile represents A B C D

5

0.67 0.75 1 1.5

In a pie chart, if wages are represented by 90° and the total cost is £550,000, what is the amount paid out in wages? A B C D

4

classes are the same width the end limits are unambiguously defined the information is grouped the number of classes is large

the mid-point the lower class boundaries the upper class boundaries any of the above

A frequency distribution of a sample of monthly incomes is as follows £ 400 and less than 800 800 and less than 1,000 1,000 and less than 1,200 1,200 and less than 1,300 1,300 and less than 1,400

Frequency 7 16 28 21 8 80

38 Exam Practice Kit: Business Mathematics If the area between £800 and £1,000 has a height of 8 cm, what is the height of the rectangle 1,000 and less than 1,200? A B C D

10 12 14 16

8 The top 1% of the population is referred to statistically as A B C D

percentile decile quartile 2nd decile

9 Which of the following are types of bar chart? (i) (ii) (iii) (iv) A B C D

simple multiple component compound i and ii i, ii, iv i, ii, iii i, ii, iii, iv

10 A histogram uses a set of rectangles to represent a grouped frequency table. To be correctly presented, the histogram must show the relationship of the rectangles to the frequencies by reference to the A B C D

height of each rectangle area of each rectangle width of each rectangle diagonal of each rectangle


The number of classes is small so odd one out is D

2

Since we have multiplied one side by 1.5 we need to divide the other by 1.5 so 0.67. so A

3

There are 360 degrees in a circle 90 so £550,000 £137,500 360 so B

4

The upper quartile represents 75% through the cumulative frequencies. so B

5

The decile represents 10% of the population. so B

Presentation of Data 39 6 Cumulative frequencies are plotted against the upper class boundaries. so C 7 Scale 1 cm for 2 frequencies, then 28 should have a height of 14. so C 8 The top 1% of the population is referred to statistically as the percentile. so A 9 Simple, multiple, component and compound are all types of bar charts. so D 10 To be correctly presented, the histogram must show the relationship of the rectangles to the frequencies by reference to the area. so B

8

Averages


Distinguish between (i) the mean (ii) the median (iii) the mode

2

State three advantages and two disadvantages of using the mean Advantages (i) (ii) (iii)

Disadvantages (i) (ii)

3

Calculate the arithmetic mean of 3, 6, 7, 8, 9, 11, 13, 15.

4

State four advantages and three disadvantages of using the median Advantages (i) (ii) (iii) (iv)

Disadvantages (i) (ii) (iii)

5

Calculate the median of 3, 6, 10, 14, 17, 19 and 22.

6

State four advantages and three disadvantages of using the mode Advantages (i) (ii) (iii) (iv)

7

Disadvantages (i) (ii) (iii)

In one over a batsman scored 4, 4, 2, 1, 0 and 4. Calculate the mode. 40

Averages 8

In his last two 72 hole competitions, a golfer scored 67, 71, 72, 73, 72, 69, 71, 72. Calculate (i) (ii) (iii)

9 10

41

his mean score his median score his mode score

How can the mode be determined from a histogram? There are 100 packets of biscuits in a box with the following weights and frequencies Weights

Frequency

100 and less than 110 110 and less than 120 120 and less than 130 130 and less than 140 140 and less than 150 150 and less than 160 160 and less than 170 170 and less than 180 180 and less than 190 190 and less than 200 What is the mean weight?

1 2 5 11 21 20 17 11 6 6


(i) (ii) (iii)

2

The arithmetic mean is calculated by taking the total value of all items divided by the total number of items. The median is the value of the middle item in a distribution once all the items have been arranged in order of magnitude. The mode is the value that occurs most frequently amongst all the items in the distribution.

The mean Advantages (i) Easy to calculate and understand. (ii) All the data in the distribution is used. (iii) It can be used in more advanced mathematical statistics. Disadvantages (i) It may give undue weight or be influenced by extreme values e.g. income (ii) The value of the average may not correspond to any individual value in the distribution for example 2.2 children.

3

3 6 7 8 9 11 13 15 72 9 8 8

4

The median Advantages (i) It is not affected by extreme values. (ii) It is easy to understand.

42 Exam Practice Kit: Business Mathematics (iii) It is unaffected by unequal class intervals. (iv) It can be the value of an actual item in the distribution. Disadvantages (i) If there are only a few items it can be unrepresentative. (ii) It is unsuitable for use in mathematical tables. (iii) Data has to be arranged in order of size which is time consuming. 5

Since there are three numbers below 14 and three numbers above 14, median is equal to 14.

6

The mode Advantages (i) It is easy to understand and calculate. (ii) It is not affected by extreme values. (iii) It can be calculated even if all the values in the distribution are not known. (iv) It can be the value of an item in the distribution. Disadvantages (i) There may be no modal value or more than one may exist. (ii) It is not suitable for mathematical statistics. (iii) Data has to be arranged to ascertain which figure appears the most often.

7

The modal value is four since it appears three times.

8

Golfer’s score

67 71 72 73 72 69 71 72 8 71 (ii) Median 67, 69, 71, 71, 72, 72, 72, 73 In this example the median is found by taking the arithmetic mean of 71 and 72 so 71.5 (iii) The score which appears the most frequently is 72. (i)

9

Mean

Estimation of mode for grouped data In a grouped frequency distribution, the modal class is the class with the largest frequency. This can easily be found by observation. The value of the mode within the modal class can then be estimated from a histogram. Having located the modal class it is necessary to draw in the dotted lines shown in the following diagram.

f

M = Modal value of the variable

M

Variable

Averages

43

10 Class interval

Mid-value

Weight (grams) 100 and less than 110 110 and less than 120 120 and less than 130 130 and less than 140 140 and less than 150 150 and less than 160 160 and less than 170 170 and less than 180 180 and less than 190 190 and less than 200

x

f

fx

105 115 125 135 145 155 165 175 185 195

1 2 5 11 21 20 17 11 6 6

105 230 625 1,485 3,045 3,100 2,805 1,925 1,110 1,170

f 100

fx 15,600

Total X

Frequency

fx 15,600 156 g f 100


The arithmetic mean of 3, 6, 10, 14, 17, 19 and 22 is A B C D

2

The median of 3, 6, 10, 14, 17, 19 and 22 is A B C D

3

11 13 14 15

The mean weight of 10 parcels is 20 kg. If the individual weights in kilograms are 15, x, 22, 14, 21, 15, 20, x, 18, 27 then the value of x is A B C D

4

11 13 14 15

20 kg 24 kg 40 kg 48 kg

The mode is the value A B C D

which appears with the highest frequency which is the same as the arithmetic mean which is the mid-point value none of the above

44 Exam Practice Kit: Business Mathematics 5 A factory employs staff in four departments for which the average mean wage per employee per week is as follows Department Mean wage No. of employees

W £50 20

X £100 5

Y £70 10

Z £80 5

The average mean wage per employee A B C D

£60 £65 £70 £75

6 If there are n items in the distribution the value of the median is n1 2 n1 B 2 C n1 D n1 A

Questions 7–9 are based on the following data A sample of 12 packets of crisps taken from a box had the following weights in grams 504, 506, 501, 505, 507, 506, 504, 508, 503, 505, 502, 504. 7 Calculate the mean weight. A B C D

502.3 503.4 504.6 505.7

8 Calculate the median weight. A B C D

504 504.5 505 505.5

9 Calculate the modal weight. A B C D 10

504 505 506 507

Which of the following is not an advantage of the median? A B C D

It is simple to understand It is not affected by extreme values It can be the value of an actual item in the distribution It is suitable for use in mathematical statistics

Averages

45

Multiple choice solutions 3 6 10 14 17 19 22 7 91 13 7

1 Arithmetic mean

so B 2 Median is the value of the middle item. So there are three numbers below 14 and three numbers above. so C 3 Mean 20 kg Samples size 10 so 15 x 22 14 21 15 20 x 18 27 200 so 152 2x 200 2x 48 x 24 so B 4 The mode is the value which appears with the highest frequency. so A 5 Dept.

Mean wage

No. of employees

Total

W X Y Z

50 100 70 80

20 5 10 5 40

1,000 500 700 400 2,600

Mean wage per employee

£2,600 £65 40

so B 6 If there are n items in the distribution the value of the median is so A 7 To make calculation easier subtract 500 461576483524 so 500 12 55 500 = 504.6 12 so C 8 Arranging in numerical order we have 501, 502, 503, 504, 504, 504, 505, 505, 506, 506, 507, 508 504 505 Median 504.5 2 so B 9 It is 504 since it appears three times. so A 10 The median is not suitable for mathematical statistics. so D

n1 . 2

9

Variation


Calculate the standard deviation of 3, 4, 6, 8, 9.

2

In last Saturday’s football matches there were 40 games played and the information below shows the number of bookings. Number of bookings 1 2 3 4 5

Frequency 3 5 12 14 6

Calculate the standard deviation. 3

Given the following data on Product A and Product B, which has the highest coefficient of variation?

Product A Product B 4

Mean

Standard deviation

5.46 16.38

1.29 4.21

What is the relationship between the mean, median and mode in (i) (ii) (iii)

a normal distribution a positively skewed distribution a negatively skewed distribution

5

If the mean is equal to 40 and the median is 37 and the standard deviation is equal to 9, calculate Pearson’s coefficient of skewness.

6

The value of sales in Jimmy Farish’s shop was January February March April May June

8,000 7,500 8,200 9,100 8,500 8,400

July August September October November December

6,200 8,100 8,200 8,100 8,400 10,000

From this data calculate the standard deviation. 46

Variation 47

Questions 7–10 are based on the following data The numbers in seconds show the lap times of 40 drivers. 126 125 128 124 127

120 127 126 127 122

122 113 117 114 106

105 112 114 111 121

129 130 120 116 116

7

Group the data into eight classes.

8

Calculate (i) (ii) (iii)

9 10

119 122 123 131 135

131 134 127 128 142

138 136 140 137 130

the median value quartile value semi-inter quartile range

Calculate the mean of this frequency distribution. Calculate the standard deviation.


x

x2

3 4 6 8 9 30

9 16 36 64 81 206

√

206 5

305

2

√41.2 36 √5.2 2.28

2

x

Frequency

fx

fx2

3 5 12 14 6 40

3 10 36 56 30 135

3 20 108 224 300 655

1 2 3 4 5 Total

√

655 40

135 40

√16.38 11.42 √4.96 2.22

2

48 Exam Practice Kit: Business Mathematics 1.29 100 23.63% 5.46 4.21 100 Product B coefficient of variation 25.7% 16.38 Product B has highest coefficient of variation.

3

Product A coefficient of variation

4

(i) (ii) (iii)

They all have the same value. Mean highest, median middle and mode lowest value. Mean lowest, median middle and mode highest value. 3(mean median) Standard deviation 3 (40 37) 9 9 9 1

5 Pearson’s coefficient of skewness

6

x2

x 8,000 7,500 8,200 9,100 8,500 8,400 6,200 8,100 8,200 8,100 8,400 10,000 98,700

√

64,000 56,250 67,240 82,810 72,250 70,560 38,440 65,610 67,240 65,610 70,560 100,000 820,570 820,570 12

98,700 12

2

√68,380.83 67,650.63 36 √730.20 27.02

7

Time 105 110 110 115 115 120 120 125 125 130 130 135 135 140 140 145

Tally II IIII III

IIII II

Frequency 2 5 4 8 10 5 4 2

Variation 49 8

By constructing cumulative frequency distribution (i) (ii) (iii)

9

125 seconds 119 seconds 131 seconds 1 semi inter quartile range (131 119) 2 6 seconds median 1st quartile 3rd quartile

Time 105–110 110–115 115–120 120–125 125–130 130–135 135–140 140–145

Mean

Mid-point 107.5 112.5 117.5 122.5 127.5 132.5 137.5 142.5

F

fx

fx2

2 5 4 8 10 5 4 2

215 562.5 470 980 1,275 662.5 550 285

23,112.50 63,281.25 55,225.00 120,050.00 162,562.50 87,781.25 75,625.00 40,612.50

f 40

fx 5,000

fx2 628,250.00

5,000 fx 125 seconds 40 f

√

628,250 40 9.01%

10 Standard deviation

5,000 40

2


Where the mean, median and mode have the same value, this is known as A B C D

2

The following data relate to a frequency distribution mean 34 median 32 standard deviation 12 Pearson’s coefficient of skewness is equal to A B C D

3

a normal distribution a positively skewed distribution a negatively skewed distribution they cannot have the same value

3 2 1 0.5

Several groups of invoices are being analysed. For each group the coefficient of variation has been calculated. The coefficient of variation measures A B C D

the range of values between the invoices the correlation between the invoice values the relative dispersion of the invoice values the variation between the sample mean and the true mean


The standard deviation of 3, 5, 8, 11 and 13 is A B C D

5

In a negatively skewed distribution (i) (ii) (iii) (iv) A B C D

6

i, ii i, ii, iii i, ii, iv i, ii, iii, iv

29.86 31.43 33.79 34.61

The interval between the upper quarter and the lower quarter is known as A B C D

8

the mean is below the median the mean is below the mode the median is above the mean the median is above the mode

If the standard deviation is 1.1 and the arithmetic mean is 3.5 then the coefficient of variation is equal to A B C D

7

3.69 4.25 5.41 7.62

the mean the standard deviation the mode the inter quartile range

Four products have the same mean weight of 250 grams but their standard deviates are Product A Product B Product C Product D

10 grams 15 grams 20 grams 25 grams

Which product has the highest coefficient of variation? A B C D 9

Product A Product B Product C Product D

Which of the following are advantages of the semi-inter quartile range? (i) (ii) (iii) (iv) A B C D

it is simple to understand it is not affected by extreme values it takes all values into account it can be used in mathematical statistics i, ii ii, iii i, iv iii, iv

Variation 51 10

In a negatively skewed distribution the peak will be closest to A B C D

the mean the mode the median none of the above


Where the mean, median and mode have the same value, this is known as a normal distribution. so A

2

Pearson’s coefficient of skewness

3(mean median) Standard deviation 3(34 32) 12 0.5

so D 3 4

The coefficient of variation measures the relative dispersion of the given data. so C x

x2

3 5 8 11 13

9 25 64 121 169

40

x2 388

√388 (40)2 √5 (5)2 √77.6 64 √13.6

r

3.69 so A 5

In a negatively skewed distribution the mean is below the median, the mean is below the mode, and the median is above the mean. so B A negatively skewed distribution f

Mean Mode Median

Variable

52 Exam Practice Kit: Business Mathematics Standard deviation 100 Arithmetic mean 110 31.43 3.5

6 Coefficient of variation

so B 7

The interval between the upper quarter and the lower quarter is known as the inter quartile range. so D

8

Product A

9

The only advantages are

10 100 4 250 15 Product B 100 6 250 20 Product C 100 8 200 25 Product D 100 10 250 so highest is Product D, so D

(i) (ii)

it is simple to understand and it is not affected by extreme values

so A 10

A distribution will peak at the mode whether it is normal, positive or negative since it is the value which occurs the most frequently, so B

10

Index Numbers


A commodity costs £105 in year 2004. In 2005 it cost £120. How much had the price risen over the year?

2

A factory produced 7,500 items in year 2004. In 2005 it had fallen to 7,000. How much did quantity fall by during the year?

Questions 3 and 4 are based on the following data Product A B

2004 price

2005 price

Quantity

£2 £15

£3 £16

2 5

3

Construct a price index using quantity weights.

4

Construct a price index using value weights.

Questions 5 and 6 are based on the following data A production process uses 10 units of labour on Product A and 30 units of labour on Product B. Costs are as follows Item

2004

2005

Product A Product B

£6.50 £5

£7 £5.20

5

Construct a price index using quantity weights.

6

Construct a price index using value weights. 53


Questions 7 and 8 are based on the following data Item

Price 2004

Quantity

Price 2005

Quantity

Milk Bread Soap Sugar Eggs

20p 40p 45p 60p 80p

50,000 pints 30,000 loaves 20,000 bars 10,000 kilos 3,000 dozen

26p 45p 50p 65p 70p

70,000 pints 40,000 loaves 25,000 bars 8,000 kilos 2,500 dozen

7 Calculate price increase between 2004 and 2005 using Laspeyre index. 8 Calculate price increase between 2004 and 2005 using Paasche index. 9

(i) (ii) (iii) (iv)

State two advantages of Laspeyre index. State two disadvantages of Laspeyre index. State two advantages of Paasche index. State two disadvantages of Paasche index.

10 The price index for a commodity in the current year is 135 (base year is equal to 100). The commodity is currently priced at £55. In the base year it was £30. (i) Convert current price to base year price. (ii) Convert base year price into current price.


120 100 14.2% 105

2

7,000 100 93.33 so fall of 7% 7,500 Item

3

Year 0

Year 1

Q

QW

VW

A B

2 15

3 16

2 5

2 5 WA7

4 75 WB79

Item

n1 100 n0

A B

150 106.7

WA

300 533.5 833.5

WA

n1 100 833.5 n0

WB

n1 100 8,602.5 n0

Quantity weights

n1 100 n0

833.5 119.1 7

WB

n1 100 n0 600 8,002.50 8,602.50

Index Numbers 55 4

Value weights

8,602.5 108.9 79

Item

P0 2004

P0 2005

Q

QW

VW

6.50 5.00

7 5.20

10 30

10 30 40

65 150 215

n1 100 n0

WA

Product A Product B Item Product A Product B

108 104

5

4,200 105 40

6

22,620 105.21 215

7

Laspeyre index

p1 100 p0

7,020 15,600 22,620

Weight Q0

Price P0

Q0 P0

P1

Q0 P1


50,000 30,000 20,000 10,000 3,000

20 40 45 60 80

10,000 12,000 9,000 6,000 2,400 39,400

26 45 50 65 70

13,000 13,500 10,000 6,500 2,100 45,100

45,100 Q0P1 14 39,400 Q0P0

Paasche index Item

Weight Q1

Price P0

Q1 P0

P1

Q1 P1


70,000 40,000 25,000 8,000 2,500

20 40 45 60 80

14,000 16,000 11,250 4,800 2,000 48,050

26 45 50 65 70

18,200 18,000 12,500 5,200 1,750 55,650

Index 9

1,080 3,120 4,200

WB

Item

Index 8

n1 100 n0

55,650 Q1P1 116 48,050 Q1P0

Easier to calculate since denominator remains the same Q0P0. Each year comparable with all other years. (ii) Using out-of-date quantities. Overstates inflation. (iii) Based on current consumption patterns. Comparable with base year. (iv) Greater amount of calculation since both numerator and denominator change. Underestimates inflation. (i)


(i) (ii)

100 £55 £40.74 135 135 £30 £40.50 100


If a commodity costs £2.60 today and £3.68 in one year’s time, how much has the price risen in the intervening period? A B C D

2

A company sold 6,000 units of product X last year. This year quantity sold was 6,500. How much has quantity sold risen by? A B C D

3

10% 23% 39.2% 41.5%

6.2% 7.5% 8.3% 10.4%

The Laspeyre price index is calculated using the formula

Q0P1 Q0P0 Q1P1 B 100 Q0P0 Q0P1 C 100 Q1P0 Q0P1 D 100 Q0P1 A 100

4

The Paasche price index is calculated using the formula

Q1P1 Q1P0 Q0P1 B 100 Q0P0 Q0P1 C 100 Q1P1 Q0P0 D 100 Q1P1 A 100

5

An index linked pension of £10,000 per annum became payable on 1st January 2002 when the index of retail prices was 135.6. If the index in 2004 is 142, how much pension should the individual receive from 2004? A B C D

10,104 10,472 11,137 11,569

Index Numbers 57 6 Which of the following are advantages of the Laspeyre index? (i) (ii) (iii) (iv) A B C D

Easier to calculate since denominator remains the same. Each year comparable with all other years. Based on current consumption patterns. Comparable with base year. i, ii ii, iii ii, iv iii, iv

7 Which of the following are disadvantages of using the Paasche index? (i) (ii) (iii) (iv) A B C D

using out-of-date quantities overstates inflation greater amount of calculation since both numerator and denominator change understates inflation i, ii i, iv ii, iii iii, iv

8 Which of the following are types of index numbers? (i) (ii) (iii) (iv) A B C D

weighted indices Laspeyre indices Paasche indices chain base indices i, ii, iii ii, iii, iv i, iii, iv i, ii, iii, iv

9 A shopkeeper saw his weekly turnover rise during the first five weeks at the following rate: Week 1 Week 2 Week 3 Week 4 Week 5

£1,000 £1,100 £1,210 £1,331 £1,464

The chain base index between week 4 and week 5 was A B C D

95 100 105 110

10 The FTSE 100 index is A B C D

an index of food prices a price index of the 100 most popular brands of food an index of the top 100 plc company share prices a price index of the basket of commodities used in an average household



2

3.68 100 141.5 2.60 so rise 41.5% so D 6,500 100 108.3 6,000 so 8.3% rise so C

3 The Laspeyre price index is calculated using the formula 100

Q0P1 . Q0P0

so A 4 The Paasche price index is calculated using the formula 100

Q1P1 . Q1P0

so A 5 £10,000

142.0 £10,472 135.6

so B 6 In the Laspeyre index the advantages are that it is easier to calculate since denominator remains the same and each year is comparable with all other years. so i, ii so A 7 The disadvantages of using the Paasche index are that a greater amount of calculation is required since both numerator and denominator change and it tends to underestimate inflation. so iii, iv so D 8 Weighted indices, Laspeyre indices, Paasche indices and Chain base indices are all types of index numbers. so D 9

£1,464 100 £110 £1,331 so D

10 The Financial Times Share Index 100 is an index of the top 100 UK plc company share prices. so C

11

Probability


If a card is pulled out of a pack of playing cards at random what is the probability that it is (i) black (ii) a club (iii) an ace (iv) the ace of spades

2

Fill in the missing word (i)

Where the probability of an event is calculated by a process of logical reasoning, this is known as ........................ probability. (ii) When a situation can be repeated a number of times this is classed as ........................ probability. (iii) Where estimates are made by individuals of the relative likelihood of events occurring, this is called ........................ probability. (iv) A pictorial representation in which terms of a mathematical statement are shown by overlapping circles is known as a ........................ diagram. 3

City and United play each other two times per season. If each side has an equal chance of winning both games, is local bookmaker, Frank Green, too generous with his odds of 5–1 on both teams completing a double, that is, one of the teams wins both matches. You can have the same 5–1 bet on either team.

4

There are 500 fish in a lake. There are 200 pike, 150 perch, 100 trout and 50 salmon. Each fish has an equal chance of being caught. (i) (ii) (iii) (iv) (v)

5

What is the probability that a pike is caught first? What is the probability that a pike or a perch is caught first? What is the probability that a pike, perch or salmon is caught first? What is the probability that a perch is not caught first? What is the probability that neither a salmon nor a trout is caught first?

In a room there are 100 CIMA students. Fifty per cent are male and 50% are female. Sixty per cent are fully qualified, 40% are partly qualified. What is the probability of selecting at random a student who is (i) (ii)

male? fully qualified? 59

60 Exam Practice Kit: Business Mathematics (iii) (iv) (v)

male and qualified? male or qualified? female or qualified?

6 Twenty-five per cent of new cars of a particular model are supplied from factory X. The remainder come from factory Y. Ten per cent of factory X’s output has a major fault, 18% of factory Y’s output has the same fault. (i) (ii) (iii)

What is the probability that a car selected at random has a major defect? What is the probability it was made at factory X? What is the probability it was made at factory Y?

7 A box contains four red balls, two white balls and a yellow ball. If three balls are selected at random and there is no replacement between each selection what is the probability of selecting (i) (ii)

three of the same colour? one of each colour?

8 If we toss a fair coin two times (i) (ii) (iii) (iv)

what is the probability that first toss is a head? what is the probability that both tosses will be heads? what is the probability that neither tosses will be heads? what is the probability that a head would not appear in three consecutive tosses?

9 A travel agent keeps a stock of holiday brochures. Currently there is a total of 500 brochures in stock, as follows: 285 for European holidays, 90 for American holidays, 110 for Asian holidays and 15 for African holidays. A brochure is selected at random. Calculate the following probabilities (i) (ii) (iii) (iv)

that a European brochure is selected that an African brochure is NOT selected that neither an American nor an Asian brochure is selected that either a European or an Asian brochure is selected

10 What is a mutually exclusive event?


(i) (ii) (iii) (iv)

2

(i) (ii) (iii) (iv)

26 52 13 52 4 52 1 52

1 2 1 4 1 13 1 52

a priori empirical probability subjective probability venn

Probability 61 3

He is not too over generous In match 1 there are three possible outcomes: City win – United win – draw In match 2 there are also three possible outcomes, so between the two matches there are nine possible outcomes, so the chances of either team beating each other twice is 9 1, not good odds.

4

(i) (ii) (iii) (iv) (v)

200 in 500 or 40% 200 150 in 500 or 70% 200 150 50 in 500 or 80% 500 150 in 500 or 70% 500 (100 50) in 500 or 70%

5

(i) 50 in 100 or 50% (ii) 60 in 100 or 60% 50 60 (iii) or 30% 100 100 (iv) Someone who is not male or qualified is female and unqualified. This probability 50 40 20% 100 100 therefore male or qualified must be 100 20% 80% 50 60 (v) same as (iii) 30% 100 100

6

Where only percentage figures are given, it is easier to use an absolute number such as 1,000 and produce the following table:

Fault OK Total

Factory X

Factory Y

Total

25 225 250

135 615 750

160 840 1,000

(i) 160 in 1,000 or 16% (ii) 25 in 160 or 16% (iii) 135 in 160 or 84% 7

There is only one colour which can be selected three times – red (i) (ii)

4 3 2 24 4 0.1 7 6 5 210 35 There are six possible outcomes that give one of each colour

so

(i) (ii) (iii) (iv) (v) (vi)

Red Red White White Yellow Yellow

White Yellow Red Yellow Red White

Yellow White Yellow Red White Red

62 Exam Practice Kit: Business Mathematics so probability 6

2 1 4 7 6 5

48 210 8 0.23 15

8

(i) Probability that first toss is head 50 50 or 1 in 2 (ii) Probability 1 1 1 in 4 2 2 (iii) If neither toss is a head then both must be a tail. Probability of two tails 1 1 1 in 4 2 2 so chances that neither are heads 1 1 in 4 3 in 4 (iv) For head not to appear in three consecutive tosses, we would need 3 tails which 1 1 1 1 is 2 2 2 8 1 7 so probability 1 8 8 (i) Probability that a European brochure is selected

9

285 0.57 500

(ii) Probability that an African brochure is not selected 485 500 15 0.97 500 500 (iii) Probability that neither an American nor an Asian brochure is selected 300 500 90 110 0.60 500 500 (iv) Probability that either a European or an Asian brochure is selected 285 110 395 0.79 500 500 10 Mutually exclusive events Two or more events are said to be mutually exclusive if the occurrence of any one of them precludes the occurrence of all others, that is only one thing can happen. For example if we throw a coin and it lands heads it cannot be a tail.

Multiple choice questions Questions 1–4 relate to the following information A pack of cards consists of 52 playing cards. 1

What is the probability that a card selected at random is the ace of hearts? A B C D

1 in 2 1 in 13 1 in 26 1 in 52

Probability 63 2

What is the probability that a card selected at random is an ace? A B C D

3

What is the probability that a card selected at random is a heart? A B C D

4

1 in 2 1 in 13 1 in 26 1 in 52

1 in 2 1 in 4 1 in 9 1 in 13

What is the probability that a card selected at random is red? A B C D

1 in 2 1 in 3 1 in 4 1 in 5

Questions 5–8 relate to a number of CIMA students who recently sat for Paper 3c

Type of student Male Female

5

1,000 500

500 300

1 in 2 1 in 3 1 in 4 1 in 5

If a student is selected at random what is the probability that they failed? A B C D

7

Total number of passes

If a student is selected at random what is the probability she is female? A B C D

6

Total number of scripts

1 in 2 1 in 3 7 in 15 8 in 15

If a student is selected at random what is the probability that the student is male or someone who failed? A B C D

0.5 0.6 0.7 0.8


If a student is selected at random what is the probability of selecting a male who failed? A B C D

1 in 2 1 in 3 1 in 4 1 in 5

Questions 9 and 10 are based on the following A box contains four red balls, two blue balls and a yellow ball. If three balls are selected at random and there is no replacement between each selection. 9 What is the probability of selecting one of each colour? A B C D

0.21 0.23 0.25 0.27

10 What is the probability of selecting three of the same colour? A B C D

0.07 0.09 0.11 0.15


There is only one such card in the pack so probability is 1 in 52. so D

2

There are four aces in a pack so

4 1 . 52 13

so B 3

There are 13 hearts in a pack of 52 so

1 13 or . 52 4

so B 4

There are 26 red cards in a pack of 52 so

26 1 or . 52 2

so A 5

Total number of females 500 Total number of students 1,500 so

1 500 . 1,500 3

so B 6

Total number of failures Total number of students so C

700 so 7 in 15. 1,500

Probability 65 7

Turn question round Someone who is not male and failed is a female who passed Total number of females who passed Total number of students 300 1 in 5 1,500 Therefore the opposite 4 in 5 or 0.8 so D

8

9

1 1,000 500 2 1 2 or 1,500 1,000 3 2 6 3 so 1 in 3 so B There are six possible ways that give one of each colour Ball 1

Ball 2

Ball 3

Red Red Yellow Yellow Blue Blue

Blue Yellow Red Blue Red Yellow

Yellow Blue Blue Red Yellow Red

So probability of one of each colour 6

2 1 8 4 0.23 7 6 5 38

so B 10

Only one colour can be selected three times – red 4 3 2 4 so chance of 3 reds 0.11 7 6 5 35 so C

Expected Value and Decision-Making

12


The local council are considering purchasing a snow plough which would cost £20,000 per annum. This would save on outside contractors but the amount would depend on the severity of the winter. Winter severe average mild

Annual savings

Probability

£40,000 £20,000 £10,000

0.2 0.5 0.3

Should the council buy a snow plough? 2

A market trader has the choice of selling umbrellas or ice cream. It has a 60% chance of raining and a 40% chance of being fair. If it rains he will make £200 profit on umbrellas and lose £50 if he chooses ice cream. If it is fair, he will lose £10 selling umbrellas and make £150 selling ice cream. Which product would you advise him to sell?

3

A new product is expected to generate the following profits: Level of demand high medium low

Profits

Probability

£100,000 £50,000 £20,000 loss

0.1 0.5 0.4

(i) What is the expected profit from the new product? (ii) What is the maximum you would invest in this product? 4

Expected values A State two advantages of expected values. B State two disadvantages of expected values.

5

Miss Jones is a supply teacher. Each day she must decide to travel to the agency where there is a 75% chance that she will get work. Her train fare is £10 return. There are three types of jobs available: (i) teaching CIMA students for £200 per day (ii) teaching ACCA students for £150 per day (iii) teaching Chartered students for £100 per day 66

Expected Value and Decision-Making 67 The chances of CIMA are 10%, ACCA 40% and Chartered 50%. Draw a decision tree which shows this information and advise Miss Jones what she should do. 6

A company is deciding between three projects A, B and C. The expected profit from each one is as follows: Profit

Project A Probability

£5,000 £2,500

0.5 0.5

Profit

Project B Probability

£10,000 £1,000

0.3 0.7

Profit

Project C Probability

£6,000 £4,000

0.4 0.6

Rank projects in descending order, stating expected values of each. 1 2 3 7

In a forthcoming sales promotion each pack of cigarettes is to contain a leaflet with eight ‘scratch off’ square patches, randomly arranged. The purchaser will scratch off one patch to reveal the value of a small prize. The value of the eight patches on the leaflet is to be as follows: Value of prize Number of patches

£0.20 5

£0.50 2

£1 1

The company has to decide on the number of packs in which to put leaflets, given a budget of £75,000. Find the ‘average cost’ of a leaflet, and deduce the number of leaflets you would use and why. 8

In another promotion for cigarettes, a leaflet pictures a roulette wheel with 37 numbers, seven of which are randomly arranged winning numbers. The purchaser is allowed to scratch off seven of the 37 numbers in the hope of winning a prize. It is therefore possible to select 0, 1, 2, 3, 4, 5, 6 or 7 winning numbers on each leaflet. (i) What is the probability of a purchase not winning a prize? (ii) If there are one million purchases during the promotion, what are the chances of the ‘Super Prize’ (the Super Prize is when all seven selections are winners) being won?

9

A golf club has to decide how many programmes to produce for a Charity Pro-Am golf tournament. From previous experience of similar tournaments, it is expected that the probability of sales will be as follows: Number of programmes demanded 1,000 2,000 3,000 4,000 5,000

Probability of demand 0.1 0.4 0.2 0.2 0.1

68 Exam Practice Kit: Business Mathematics The best quotation from a local printer is £2,000 plus 10 pence per copy. Advertising revenue totals £1,500. Programmes are sold for 60 pence each. Unsold programmes are worthless. Draw up a profit table with programme production levels as columns and programme demand levels as rows. 10 Find the most profitable number of programmes to produce; to explain your analysis, including any assumptions.


Cost of new machine £20,000 Expected saving (0.2 40,000) (0.5 20,000) (0.3 10,000) £8,000 £10,000 £3,000 £21,000 Based purely on expected value theory, the council should buy a new snow plough since they could save £1,000 more than the machine costs.

2

0.6 Rain

Loss £50

0.4 Fair

Profit £150

Rain 0.6

Profit £200

Ice cream

Umbrellas Fair 0.4

Loss £10

Expected value selling ice cream (0.6 £50) 0.4 (£150) £30 £60 £30 Expected value selling umbrellas (0.6 £200) (0.4 £10) £120 (£4) £116 so choose umbrellas. 3

(i)

Expected profit from new product is (0.1 £100,000) (0.5 £50,000) (0.4 20,000) £10,000 £25,000 £8,000 £27,000 (ii) £27,000

4

A

(i) It is an objective way of making investment decisions. (ii) Over the long term, the correct decision will be taken.

B

(i) On individual projects the wrong decision may be made because of random events. (ii) A 10% chance of winning £1,000 would show a higher expected value than a 90% chance of winning £100 but would it be the correct decision to take.

Expected Value and Decision-Making 69 5

CIMA (10%) £200 Work available

ACCA (40%) £150 Chartered (50%) £100

Look for work costs £10

25% No work available

Stay at home

If there is work she can expect 10% of £200 40% of £150 50% of £100 £20 £60 £50 £130 That is only if work is available, so expected value 75% £130 25% 0 £97.50 However, whether she finds work or not she will still have to pay £10 for train fare so her expected income from work £97.50 £10 £87.50 If she stays at home her expected value is £0 so advise Miss Jones to seek work. 6

Project A (0.5 £5,000) (0.5 £2,500) £3,750 Project B (0.3 £10,000) (0.7 £1,000) £3,700 Project C (0.4 £6,000) (0.6 £4,000) £4,800 so 1 C 2A 3B

7 Cigarette sales promotion The average value of a prize is calculated as follows Value (x) £ 0.20 0.50 1.00

Frequency f

5 2 1 8 3.00 fx 37.5p Mean 8 f

fx 1.00 1.00 1.00 3.00

Total cost Average cost per leaflet no. of leaflets £75,000 0.375 x £75,000 x 200,000 0.375 So 200,000 leaflets should be used.


30 (i) Probability that first number uncovered is not a winner 37 This would leave 36 numbers and 29 non-winners. 29 Probability that the second number is not a winner 36 This would leave 35 numbers and 28 non-winners. Similarly for all 7 numbers, hence 29 28 27 26 25 24 30 P (no win) 37 36 35 34 33 32 31 0.1977 7 (ii) Probability that first number uncovered is a winner 37 This would leave 36 numbers and 6 winners. 6 Probability that the second number is a winner etc. 36 Hence 7 6 5 4 3 2 1 P (all win) 37 36 35 34 33 32 31 9.713 108 The expected number of Super Prizes in 1 million cards is 9.713 108 106 (i.e. probability no. of cards) 0.09713 This is the probability that the Super Prizes is won in one promotion. 1 The chance is, therefore 1 in 0.09713 1 in 10 (approximately)

9

Contribution per programme sold is (60p 10p) 50p. The table shows expected profits (needed for part (b)) found by working 4. Demand (probability in brackets)

Production 3,000

4,000

5,000

1,000

2,000

1,000 (0.1) 2,000 (0.4) 3,000 (0.2) 4,000 (0.2) 5,000 (0.1)

(W1) £0 £0 £0 £0 £0

(W2) £(100) (W3) £500 £500 £500 £500

£(200) £400 £1,000 £1,000 £1,000

£(300) £300 £900 £1,500 £1,500

£(400) £200 £800 £1,400 £2,000

Expected £ value

£0

(W4) £440

£640

£720

£680

Workings £ (W1) Contribution from sales of 1,000 Less: (Fixed print cost Advertising revenue)

500 (500) nil

(W2) Contribution from sales of 1,000 Less Less: 1,000 programmes printed and sold @ £0.10

500 (500) (100) (100)

Expected Value and Decision-Making 71 (W3) Contribution from sales of 2,000 Less

1,000 (500) 500

(W4) (0.1 100) (0.4 500) (0.2 500) (0.2 500) (0.1 500) £440. 10

Optimal production level Note: this uses the expected values found in (a). The production level with the highest expected profit is 4,000 copies.


A retailer has the choice of selling Product A which has a 0.4 chance of high sales and a 0.6 chance of low sales. High sales would yield a profit of £600. Low sales would yield a profit of £100. If Product B was sold there is a 0.6 chance of high sales and a 0.4 chance of low sales which would result in a profit of £400 or a loss of £50. Which product would be chosen and what would be the expected value? A B C D

2

Product A Product B Product A Product B

£300 £220 £220 £300

A decision tree is A a way of applying expected value criterion to situations where a number of decisions are made sequentially B a random outcome point C a statistical device used purely for retailers D none of the above

3

The difference between a decision point and a random outcome point is A decision points are where the decision made has no control over destiny B a decision point and a random outcome point are the same C a decision point is chosen by the decision maker, at a random point it is outside the control of the decision maker D none of the above

4

Ten per cent of golf balls have a minor defect. They are packaged in boxes of six. What is the probability that a box selected at random has no defects? A B C D

5

0.41 0.45 0.51 0.53

If the three possible outcomes of a decision are profits of £10, £50 and £80 with probabilities of 0.3, 0.3 and 0.4, what is the expected profit? A B C D

£40 £44 £47 £50


A supermarket is opening a new store and they have identified two sites A and B with 0.8 chance of making £400,000 profit per annum and a 0.2 chance of incurring an £80,000 loss. The expected value of those sites is A B C D

£300,000 £302,000 £304,000 £306,000

7 A newspaper vendor buys daily newspapers each day which have a resale value at the end of the day of zero. He buys the papers for 15p and sells them for 30p. The levels of demand per day and their associated probabilities are as follows Demand per day 400 440 480 520

Probability 0.2 0.3 0.4 0.1

How many newspapers should the vendor buy each day? A B C D

400 440 480 520

8 A social club has a lottery draw based on numbers between 1 and 40. If they pay £50 for the winning number and there are no other expenses how much will they need to sell each ticket for in order to make £50 profit? A B C D

£2 £2.50 £3 not enough information given

9 If a roulette table has 37 numbers 0–36 and pays odds of 35–1 on punters guessing the correct number, what is the expected rate of return on a £100 investment? A B C D

5% 5% 95% 100%

10 A new car is worth £20,000. The probability of this car being stolen or being involved in an accident over a years is 1%. The driver pays an insurance policy of £500 per annum. The annual expected value to the insurance company is A B C D

£500 £400 £300 £200

Expected Value and Decision-Making 73


Product A expected value 0.4 £600 0.6 £100 £240 £60 £300 Product B expected value 0.6 £400 0.4 £50 £240 £20 £220 Product A with higher expected value so A

2

A decision tree is a way of applying expected value criterion to situations where a number of decisions are made sequentially. so A

3

A decision point is where the decision maker can make a choice. A random outcome point is outside the control of the decision maker. so C

4

Probability of having 0 defects is

(9)6 = 0.53 10

so D 5

0.3 0.3 0.4 £10 £50 £80

£3 £15 £32 £50

so D 6

0.8 0.2 £320,000 £1,600 £400,000 £80,000 £304,000 so C

7

If the newsagent purchased and sold 400 papers per day, his profit would be £60 (400 15p). If the newsagent buys 400 but demand is 440 profit is still only £60 because he has only sold 400. If the newsagent buys 440 the cost is £66 but if he only sells 400 30p his profit is reduced to £120 £66 £54. So a pay off table would look like Demand 400 400 440 480 520

£60 £60 £60 £60

Purchased per day 440 480 £54 £66 £66 £66

£48 £54 £72 £72

520 £42 £54 £66 £78

74 Exam Practice Kit: Business Mathematics To calculate the expected value Demand per day 400 440 480 520

Probability 0.2 0.3 0.4 0.1

400

Purchased per day 440 480

520

12 18 24 6 60

10.8 19.8 26.4 6.6 63.60

8.4 16.2 26.4 7.8 58.8

9.6 16.2 28.8 7.2 61.8

Highest expected value 440 £63.60 so B 8 Amount required £50 winning number £50 profit £100 No of tickets available 40 so £2.50 each so B 9

35 100 95% 37 So on a £100 stake our rate of return is 5% so A

10 The driver will pay a premium of £500 whether there is an accident or not so expected value premium of £500 less 1% of £20,000 (£200) £300 so C

The Normal Distribution

13


State five features of a normal distribution curve. (i) (ii) (iii) (iv) (v)

2

Give three examples where a normal distribution might appear in real life. (i) (ii) (iii)

3

A group of workers have a weekly wage which is normally distributed with a mean of £400 and a standard deviation of £60. What is the probability of a worker earning. (i) (ii) (iii) (iv) (v)

more than £430 less than £350 more than £460 between £350 and £430 between £430 and £460?

4 Confidence limits (i) (ii) (iii) 5

There is a 95% probability that the population mean lies within / .............. standard errors of a sample mean. There is a 98% probability that the population mean lies within / .............. standard errors of a sample mean. There is a 99% probability that the population mean lies within / .............. standard errors of a sample mean.

The Island of Dreams has a temperature which is normally distributed with a mean of 70° and a standard deviation of 5°. What is the probability of A

(i) the temperature below 60° (ii) no lower than 65° (iii) higher than 85° (iv) between 67° and 74° B What is the maximum temperature that has no more than a 1% chance of being exceeded? 75


The mean weight of a bag of crisps is 50 g and a standard deviation of 10 g. What is the probability that a sample of 100 packets will have a mean of less than 48 g?

7

A sample of petrol invoices are selected at random. The average invoice was £47.50 with a standard deviation of £6.30. What are the 95% confidence limits for the average of all invoices?

8

Recent results have showed that 50% of CIMA students pass their foundation papers at the first attempt. What is the probability that 55% or more of a group of 250 students will pass first time? During an internal audit, an accountant had to sample a very large batch of invoices for value and for errors. A simple random sample of 200 invoices revealed a mean value of £90 with a standard deviation of £40.

9 10

(i) Find the standard error of the mean. (ii) Find 99% confidence limits for the mean value of the whole batch. (i)

Find 95% confidence limits for the error rate of the whole batch if 20 invoices showed errors. (ii) Find the size of sample required in 9(ii) and 10(i) in order to double the accuracy of your answers.


(i) (ii) (iii) (iv)

2

(i) heights of people (ii) weights of people (iii) examination marks

3

It is symmetrical and bell shaped. Both tails approach but never reach the X axis. The mean, median and mode are equal. The area under the curve is equal to 1 and the areas left and right of the mean are equal to 0.5 each. (v) It is a mathematical curve which closely fits many natural occurring distributions.

(i)

(ii)

(iii) (iv)

(v)

430 400 0.5 60 from table area above 430 is 0.3085 or approximately 31%. 350 400 0.83 60 from table area below 350 is 0.2033 or approximately 20%. 460 400 1 60 from table area above 460 0.1587 or approximately 16%. Area less than £350 has already been found in (ii) 0.2033 therefore area between £350 and £400 0.5 0.2033 0.2967 similarly area over £430 has already been found in (i) 0.3085 therefore area between £400 and £430 0.5 0.3085 0.1915 therefore area between £350 and £430 0.2967 0.1915 0.4882 so approximately 49% earn between £350 and £430. If area over £430 is 0.3085 then area between £400 and £430 is 0.5 0.3085 0.1915 If area over £460 is 0.1587 then area between £400 and £460 is 0.5 0.1587 0.3413 so area between £430 and £460 0.3413 0.1915 0.1498 so approximately 15% earn between £430 and £460.

The Normal Distribution 77 4

(i) 1.96 (ii) 2.33 (iii) 2.58

5

A

B

6

x v 60 70 10 2 5 5 from tables 0.02275 Thus the chance of temperature being below 60° is 2.2%. 65 70 (ii) Z 1 5 from tables probability of temperature less than 65° is 0.1587. So probability that temperature is no lower than 65% 1 0.1587 0.8413 So there is an 84% chance temperature is no lower than 65°. 85 70 (iii) Z 3 5 Since most normal distribution tables only go up to 2.99 we cannot find value of 3. However, taking a guess it would be about 1%. (i) Z

u 50 10 n 100 x 48

10 0.1 n 100 48 50 Z 2 1 from table area 0.2275 that is 48 is Z standard errors below mean weight of 50 so probability 22.75%.

Standard error

7

x £47.50 SD £6.30 n 40 6.30 SD n 40 95% confidence limit £47.50 / 1.96 £47.50 / 1.95 we can be 95% confident that the average invoice lies between £45.55 and £49.45.

Standard error

8

The population proportion is 50% 0.5 p 0.5 1 p 0.5 n 250 Standard error

√ √

p (1 p) n 0.5 0.5 250

√0.001 0.0316

78 Exam Practice Kit: Business Mathematics Sample proportion 0.55 0.55 0.50 1.58 So Z 0.0316 from table 1.58 0.0571 So the chances of 55% or more passing is 0.0571, approximately 5.7%. 9 Batch of invoices (i)

Standard error of the mean S £40 n 200 x 90 S SE √n 40 √200 £2.83

(ii) u x Z SE For 99% confidence limits Z 2.58 u 90 2.58 2.83 90 £7.30 lower limit £82.70 upper limit £97.30 10

Proportion of invoices containing errors: 20 0.10 200 q1p 1 0.1 q 0.9 p Z SE For 95% confidence, Z 1.96

(i) p

SE

√

pq n

√0.1 0.9 200

0.3 0.0015 200 SE 0.0212 0.10 1.96 0.0212 0.10 0.0416 Therefore lower limit 0.10 0.0416 0.584 i.e. 5.84% upper limit 0.10 0.0416 0.1416 i.e. 14.16% Both standard error formula contain √n as a denominator. (ii) Therefore if the sample size is quadrupled the range will be halved, and the accuracy doubled. √4n 2√n

The Normal Distribution 79

Multiple choice questions Questions 1–7 are based on the following information A group of workers have a weekly wage which is normally distributed with a mean of £360 per week and a standard deviation of £15. 1

What is the probability a worker earns more than £380? A B C D

2

What is the probability a worker earns less than £330? A B C D

3

1% 15% 20% 25%

What are the limits which enclose the middle 95%? A B C D

7

50% 75% 90% 95%

What is the probability a worker earns between £370 and £400? A B C D

6

5% 4% 2% 0%

What is the probability a worker earns between £330 and £390? A B C D

5

1% 2% 3% 4%

What is the probability a worker earns more than £420? A B C D

4

4% 5% 7% 9%

£330.60 and £389.40 £325.05 and £394.95 £320.40 and £400.40 £300 and £450

What are the limits which enclose the middle 98%? A B C D

£330.60 and £389.40 £325.05 and £394.95 £320.40 and £400.40 £300 and £450

80 Exam Practice Kit: Business Mathematics 8 A normal distribution has a mean of 75 and a variance of 25. The upper quartile of this distribution is A B C D

58.25 71.65 78.35 91.75

9 A normal distribution has a mean of 150 and a standard deviation of 20. Eighty per cent of this distribution is below A B C D

150 154.8 159.6 166.8

10 In a normal distribution with a mean of 150, 6.68% of the population is above 180. The standard deviation of the distribution is A B C D

10 15 20 25

Multiple choice solutions 380 360 1.33 15 From normal distribution table Z 0.4082 So probability Z 0.5 0.4082 0.0918 9% so D

1 Z

330 360 2 15 From normal distribution table Z 0.4772 So probability Z £330 0.5 0.4772 0.0228 2% so B

2 Z

3

420 360 4 15 Highest value in the normal distribution table is 3.5, so it is impossible for worker to earn more than £420 from the data so no chance 0%. so D Z

The Normal Distribution 81 4 Area between 330 and 360 0.4772 from Question 2. 390 360 2 is also 0.4772 15 so 0.4722 0.4722 0.9544 so approximately 95%. so D 5 Required area is between 370 and 400 400 360 2.666 0.4962 15 370 360 0.67 0.2486 15 so 0.4962 0.2486 0.2476 or approximately 25%. so D 6 x1 360 1.96 15 £389.40 x2 360 1.96 15 £330.60 so middle 95% lies between £330.60 and £389.40. so A 7 For an area of 0.49 Z 2.33 x 360 2.33 15 £360 (£15 2.33) £394.95 £360 (£15 2.33) £325.05 Middle 98% lies between £325.05 and £394.95. so B 8 The area under the normal curve we require the Z factor to be 0.25. Combined with the area below the mean, this will give a total value of 0.75 that is, the upper quartile Nearest value to 0.25 0.2486 0.67 So 0.67 standard deviation √25 5 We get 3.35. The upper quartile is 3.35 above the mean of 75 so 78.35. so C 9 From the normal distribution table, 30% of a distribution lies between the mean and 0.84 standard deviation above the mean so x 150 (0.84 20) 166.8 so D 10 If 6.68% of the population is above 180, then 0.5 0.0668 0.4332 so Z 1.5 x1 u 1.5 so Z u 1.5 180 150 30 u 1.5 20 so C

14

Statistical Inference


95% confidence limits x 98% confidence limits x 99% confidence limits x

........ standard errors ........ standard errors ........ standard errors

2

In a sample of 100 students, their mean height was 168.75 cm and the standard deviation was 7.5 cm. Find the 95% confidence intervals.

3

A sample of 100 items in a production line had a mean weight of 8.4 kg with a standard deviation of 0.5 kg. What is the 95% confidence interval for the mean weight of all items on the production line?

4

Past experience shows that 50% of students pass Business Mathematics at the first attempt. What is the probability that 55% or more of a group of 200 students pass?

5

In a random sample of 100 voters 55% stated they would vote for candidate A. Calculate the 95% confidence limits for the proportion of all voters in favour of candidate A.

Questions 6, 7 and 8 are based on the following data A random sample of 100 candles was found to have a mean life of 360 hours with a standard deviation of 30 hours. 6

Calculate the standard error of the mean.

7

Calculate (i) the 95% confidence intervals (ii) the 99% confidence intervals

8

The sample size necessary to provide a degree of accuracy within ................ hours at the 95% level.

Questions 9 and 10 are based on the following information In a random sample of 400 of a direct mail CD club, the mean value per order was £30 with a standard deviation of £10. The average price per CD was £15. 82

Statistical Inference 83 9 Find 95% confidence limits for the mean and what does this indicate? 10 If the sample contains 80 teenagers, find 99% confidence interval for the population percentage of teenage customers.


(i) 1.96 (ii) 2.33 (iii) 2.58

2 95% confidence interval 168.75 1.96

3

4

7.5

√100

168.75 1.96 0.75 168.75 1.47 167.28 170.22 95% confidence interval x 1.96 √n 0.5 8.4 1.96 √100 8.4 1.96 0.05 8.302 8.498 Population proportion 50% or 0.5 p 0.5 q 1 0.5 n 200 Standard error

√ √

pq n 0.5 0.5 200

√0.00125 0.03536 Standardising 0.55 z

0.55 0.5 1.41 0.03536

From table 0.5 0.4207 0.0793 Thus chances of 55% or more passing is approximately 8%. 5

p 0.55

q 1 0.55 0.45

So standard error

√ √

n 100

pq n

0.55 0.45 100 0.04975 95% confidence limits 0.55 1.96 0.04975 0.55 0.0975 0.45 0.65

This means there is a 95% probability that the proportion of all voters in favour of candidate A is between 45% and 65%.

84 Exam Practice Kit: Business Mathematics 6 Standard error 7

√n

30

√100

3 hours

(i) 95% confidence interval for u x 1.96

√n

360 1.96 3 360 5.88 354 hours 366 hours approx. (ii) 99% confidence interval for u x 2.58 3 360 7.74 So 352.26 367.74 352 368 hours to nearest hour 8 1.96 1.96

√n 30

√n

3 3

1.96 30 √n 3

√n 19.6

n 384.16 n must be at least 385 9 Standard error

√n

£10

√400

50p

95% confidence limits £30 1.96 50p £30 98p £29.02 and £30.98 This means that there is a probability of 95% that the mean value of all orders lies between £29.02 and £30.98.

Standard error 99% confidence limits

√

p(1 p) n 80 p 0.2 400 0.2 0.8 400 0.02 sample proportion 2.58 standard errors 0.2 2.58 0.02 0.2 0.0516

10 Standard error or proportion

√

This means that there is a 99% probability that the percentage of all customers who are teenagers is between 14.84% and 25.16%.

Multiple choice questions Questions 1–4 are based on the following information A manufacturer of light bulbs needs to estimate the average burning life of each bulb he makes. From a random sample of 100 bulbs was found to have a mean life of 340 hours with a standard deviation of 30 hours.

Statistical Inference 85 1

The standard error of mean was A B C D

2

The 95% confidence interval was A B C D

3

334–346 332–348 330–350 328–352

The 99% confidence interval was A B C D

4

1 hour 2 hours 3 hours 4 hours

334–346 332–348 330–350 328–352

The sample size necessary to provide a degree of accuracy within 3 hours at the 95% level was A B C D

370 375 380 385

Questions 5 and 6 are based on the following data A sample of 40 invoices are selected at random. The average value of the sample was £48.20 with a standard deviation of £7.40. 5

The 95% confidence limits for the average of all invoices is A B C D

6

How many invoices would need to be inspected for the average to be estimated to within £2 either side? A B C D

7

£45.91–£50.49 £40.91–£55.49 £35.91–£60.49 £30.91–£65.49

51 52 53 54

A sample size 100 has a mean of 120.87 and a variance of 81. The 95% upper confidence limit of the population mean is A B C D

122.634 123.162 127.823 136.746

86 Exam Practice Kit: Business Mathematics 8 A sample standard deviation tends to underestimate the population standard deviation. √n . A better estimate is obtained by multiplying the sample standard deviation by n1 This is known as A B C D

Bessels correction Pearson’s coefficient Statistical inference Point estimate of the parameter

9 A sample of 100 items on a production line has a mean weight of 8.4 g with a standard deviation of 0.5 g. What is the 95% confidence interval for the mean weight of all items on the production line? A B C D

8.302 u 8.498 8.264 u 8.569 8.154 u 8.623 8.042 u 8.749

10 In a random sample of 100 drives a golfer has a mean drive of 340 yards with a standard deviation of 30 yards. The standard error of the mean was A B C D

3 yards 30 yards 300 yards impossible to determine

Multiple choice solutions 1 Standard error

Standard deviation 30

√ 100

√n 3 hours

so C 2

95% confidence interval x 1.96 standard error 340 1.96 3 hours 340 5.88 334–346 so A

3

99% confidence limit 340 2.58 3 340 7.74 hours 332–348 so B

4 The error in the estimate 3 hours Standard deviation 3 1.96 √n 30 1.96 3 √n

Statistical Inference 87 1.96 30 √n 3 19.6 √n n 384.16 so n must be 385 so D 5 95% confidence interval 48.20 1.96 1.171 48.20 2.29 Thus we can be 95% confident that average invoices are between £45.91 and £50.49. so A 6 1.96

7.40

√n

Rearrange

2 1.96 (7.40) √n n (7.252)2 53

so C 7 The 95% confidence interval for the population mean is x 1.96 √n where x is the mean is the standard deviation n is the sample size We are given a variance of 81, the standard deviation √variance, so Standard deviation () √81 9 The given values and the calculated standard deviation need only be entered into the expression. (9) Hence 120.87 1.96 (√100) 120.87 1.96(0.9) 120.87 1.764 The question asks for the upper confidence limit 120.87 1.764 122.634 so A 8 This is known as Bessels Correction so A 9 95% confidence interval x 1.96 √n 0.5 8.4 1.96 √100 g 8.4 1.96 0.05g 8.302 u 8.498 so A 10 Standard error of mean

30

√100

so A

3 yards

√n

Correlation and Regression

15

Concepts, definitions and short-form questions Questions 1 and 2 are based on the following Records have been kept over eight quarters of the power costs of a central heating system and the hours used, as follows Period

Hours used

1 2 3 4 5 6 7 8

Power costs

25 22 16 12 7 8 15 12

124 131 98 74 56 65 114 86

1

Using the method of least squares, calculate the fixed and variable elements of cost.

2

If the coefficient of determination is 0.87, what does this signify?

Questions 3 and 4 are based on the following Fertiliser (kg used) 100 200 300 400 500 600 700

Yield (tonnes) 40 45 50 65 70 70 80

3

What is the regression line for y on x?

4

Calculate the correlation coefficient. 88

Correlation and Regression 89 5 The following table shows the ranking of six students in two tests. Student A B C D E F

Maths test

English test

4 5 2 1 3 6

2 3 1 4 5 6

Is there any correlation between the two tests? 6

(i) for a perfect correlation R (ii) for a perfect negative correlation R (iii) for no correlation R

7 A company’s weekly costs £c were plotted against production levels (P) and a regression line calculated to be C £1,000 £7.5 g. Calculate the total cost if 5,000 units were produced. 8 Reject rates achieved by 100 factory operatives is to be found by the regression equation y 20 0.25x where y % of reject rates and x the months of experience. What would be the predicted reject rate for an operator with one year’s experience?

Questions 9 and 10 are based on the following data You are asked to investigate the relationship between what a tyre company spend on rubber and their production. You are given information over the past ten months. Month

1

2

3

4

5

6

7

8

9

10

Production X 000 units Rubber costs Y £000

30 10

20 11

10 6

60 18

40 13

25 10

13 10

50 20

44 17

28 15

9 Find the least squares regression line for rubber costs. 10 If production is budgeted for 15,000 units and 55,000 units for the next two months, how much is likely to be spent on rubber?


X 25 22 16 12 7 8 15 12 X 117

Y

XY

X2

124 131 98 74 56 65 114 86 Y 748

3,100 2,882 1,568 888 392 520 1,710 1,032 XY 12,092

625 484 256 144 49 64 225 144 X2 1,991

90 Exam Practice Kit: Business Mathematics Y a bx where a fixed costs and b variable costs (8 12,092) (117 748) b (8 1,991) (117 117) 96,736 87,516 15,928 13,689 9,220 2,239 £4.12 117 748 4.12 a 8 8 93.50 60.26 £33.24 Fixed cost £33.24 Variable cost £4.12 2

A value of 0.87 indicates a high degree of positive correlation between hours used and power costs. This tells us that 87% of the variation in power costs can be attributed to changes in the hours used and 13% on other factors. However, a sample of eight is quite small. Nevertheless, 0.87 is close to 1 which indicates a perfect positive relationship.

Solutions 3 and 4 X

Y

XY

X2

1 2 3 4 5 6 7 28

40 45 50 65 70 70 80 420

40 90 150 260 350 420 560 1,870

1 4 9 16 25 36 49 140

(7 1,870) (28 420) (7 140) (28 28) 13,090 11,760 6.79 980 784 28 420 6.79 a 7 7 60 27.16 32.84 Regression line for y on x y 32.84 6.79x

3 b

4

7 1,870 (28 420)

√7 140 (28 28) 7 26,550 420)2

13,090 11,760

√(980 784)(185,850 176,400) 1,330

√196 9,450 0.98

Correlation and Regression 91 5 Student A B C D E F

Maths test

English test

D

D2

4 5 2 1 3 6

2 3 1 4 5 6

2 2 1 3 2 0 D0

4 4 1 9 4 0 D2

6 22 6 (62 1) 132 1 210 78 210 0.37 No link

r1

6

(i) 1 (ii) 1 (iii) 0

7 £1,000 5,000 (7.5 g) £1,000 £37,500 £38,500 8 y 20 0.25x If x 12 y 20 0.25 12 y 20 3 y 17% 9 x 320 y 130 n 10 xy 4,728 10 4,728(320 130) 10 12,614 3202 5,680 23,740 0.239

b

130 0.239 320 10 5.34

a

Least squares regression is y 5.34 0.239x 10 5.34 0.239 15 8.93 so £8,930 5.34 0.239 55 18.5 so £18,500



If x 560 y 85 x2 62,500 xy 14,200 and n 12, the regression line of y on x is equal to A B C D

2

In a forecasting model based on y a bx, the intercept is £234. If the value of y is £491 and x 20 then b is equal to A B C D

3

12.25 12.85 13.35 13.95

A company’s weekly costs £C were plotted against production levels for the last 50 weeks and a regression line C 1,000 250p was found. This would denote that A B C D

4

0.281 6.03x 6.03 0.281x 0.281 6.03x 6.03 0.281x

fixed costs are £1,250 variable costs are £1,250 fixed costs are £1,000; variable costs £2.50 fixed costs are £250; variable costs are £1,000

The following table shows the ranking of six students in their CIMA Economics and CIMA Business Mathematics Student A B C D E F

Economics rank

Maths rank

4 5 2 1 3 6

2 3 1 4 5 6

What is the correlation between the two subjects? A B C D 5

0.31 0.33 0.35 0.37

Management accountants require to calculate costs. The variable to be predicted is known as the A B C D

dependent variable statistical variable independent variable high-low variable

Correlation and Regression 93 6 Electricity costs based on production levels would tend to be A B C D

perfect positive linear perfect negative linear high positive low negative

7 The coefficient of determination (122) explains the A percentage variation in the coefficient of correlation B percentage variation in the dependent variable which is explained by the independent variable C percentage variation in the independent variable which is explained by the dependent variable D extent of the casual relationship between the two variables 8 In a forecast model based on y a bx, the interest is £234. If the value of y is £491 and x is 20 then the value of the slope A B C D

24.55 12.85 12.85 24.85

9 In the equation y a bx, a is equal to A B C D

the intercept the gradient the regression line the coefficient

10 For a perfect positive correlation we want the value of R to be A B C D

greater than 1 equal to 1 equal to 0 equal to 1


Equation of line is y a bx 122,800 (12 14,200) (560 85) 12 62,500 560 560 436,400 b 0.281 85 560 a 0.281 6.03 12 12 Regression line is y 6.03 0.281x so B b

2

491 234 20b 491 234 12.85 b 20 so B

94 Exam Practice Kit: Business Mathematics 3 Fixed costs are £1,000; variable costs are £2.50 so C 4 Student A B C D E F

Economics

Maths

D

D2

4 5 2 1 3 6

2 3 1 4 5 6

2 2 1 3 2 0

4 4 1 9 4 0 22

1 6(22) 1 132 6(36 1) 210 1 0.63 0.37 so D 5 The variable to be predicted depends on some other variable so A 6 Perfect positive linear, that is, rise by a constant amount so A 7 The coefficient of determination (R2) explains the percentage variation in the dependent variable which is explained by the independent variable. so B 8 Y a bx 491 234 20b 491 234 b 12.85 20 so C 9 In the equation Y a bx, a is equal to the intercept so A 10 For a perfect positive correlation we want the value of R to be equal to 1. so B

16

Time Series


An inflation index and a sales index of a company’s sales for the last year are as follows: Quarter Sales index Inflation index

1

2

3

4

109 100

120 110

132 121

145 133

Calculate the real value of sales for quarter 4. 2

Give an example of one of the following (i) long-term trend (ii) cyclical variation (iii) seasonal variation (iv) random variation

3

In an additive time series model A T C S R, the initials stand for (i) A (ii) T (iii) C (iv) S (v) R

Questions 4–7 are based on the following information The takings (in ’000s) at Mr Li’s Takeaway for the past 16 quarters are as follows: Quarter

1

2

3

4

2000 2001 2002 2003

13 16 17 18

22 28 29 30

58 61 61 65

23 25 26 29

4

Calculate the four quarterly moving average.

5

Calculate the trend.

6

Calculate the quarterly variation. 95


If Mr Li thinks his takings for the four quarters in 2004 will be £19,000, £32,000, £65,000 and £30,000, has the upward trend continued?

8

A product has a constant trend in its sales and is subject to the following quarterly seasonal variations. Quarter Seasonality

Q1

Q2

Q3

Q4

50%

50%

50%

50%

Assuming a multiplicative model for the time series, what should sales be for quarter 3, if sales in last quarter, Q2, were 240? 9

Based on the last 18 periods, the underlying sales trend is y 345 1.5x. If the seasonal factor for period 19 is 23.5, if we assume an additive forecasting model, what is the forecast for period 19?

10

Over the past 15 months, sales have had an underlying linear trend of y 7.5 3.8x where y is the number of items sold and x is the month of sale. Month 16 is expected to be 1.12 times the trend value. What is the sales forecast for month 19?

Concepts, definitions and short-form solutions 145 100 109 same as Quarter 1 133

1

Quarter 4

2

(i) change in population (ii) down turn in economic activity (iii) rise in goods sold before Christmas (iv) events in New York September 11th 2001

3

A actual value for the period T trend component C cyclical component S seasonal component R residual component

4

1st quarter

5

Trend equals central value of four quarterly moving average So 1st trend 29 30 2nd trend 30 31 So rounded up 30, 31, 32, 33, 33, 33, 33, 33, 34, 34, 35, 36

13 22 58 23 116 4 29 2nd quarter 22 58 23 16 119 4 30 So 29, 30, 31, 34, 33, 33, 33, 33, 33, 34, 34, 35, 36

Time Series

97

16 5 29 8

6

Quarter 1 Quarter 2 Quarter 3 Quarter 4

7

19 (16) 35 32 (5) 37 65 29 36 30 (8) 38 Yes, upward trend has continued

8

Forecast T S Sales last quarter 240 Q2 Seasonally for Q2 50

∴ S 150

240 100 160 150 S 50 Seasonality 50% 160 50 Forecast 80 100 Trend

y 345 1.5x x 19 so y 345 28.5 316.5 Seasonally adjusted 316.5 23.5 293

9

y 7.5 3.8x Seasonal variation 1.12 trend For month 19 y (7.5 3.8 16) 1.12 (7.5 60.8) 1.12 68.3 1.12 y 76.5

10

Multiple choice questions Questions 1–5 are based on the following information The figures below relate to the number of daily visitors to an hotel aggregated by quarter.

2000 2001 2002 2003 1

Quarter 1

Quarter 2

Quarter 3

Quarter 4

– 90 22 10

– 120 60 80

– 200 164 192

88 28 16 –

The first figure to go in the 4th quarter total is A B C D

498 438 428 418


The first figure to go in the 8th quarter total is A B C D

3

The average seasonal variation for the first quarter in 2002 was A B C D

4

88 90 92 94

The influence of booms and slumps in an industry is a measure of A B C D

8

275 300 325 350

Over an 18 month period, sales have been found to have an underlying linear trend of y 7.112 3.949x where y is the number of items sold and x represents the month. Monthly deviations from trend have been calculated and month 19 is expected to be 1.12 times the trend value. The forecast number of items to be sold in month 19 is A B C D

7

30 40 45 50

Based on the last 15 periods, the underlying sales trend is y 345.12 1.35x. If the 16th period has a seasonal factor of 23.62, assuming an additive forecasting model, then the forecast for the period is A B C D

6

3 1 1 3

What is the expected number of daily visitors for the 4th quarter of 2003 if trend figures is 120? A B C D

5

900 918 936 950

long-term trends cyclical variations seasonal variations random variations

The series X1 X2 X3 . . . Xn can be expressed as A 3x 3 X1 Xn B i1 n

C

Xi

i1

D 3x N

Time Series 9

99

An inflation index and index numbers of a company’s sales (£) for the last year are given below. Quarter Sales (£) index Inflation index

1

2

3

4

109 100

120 110

132 121

145 133

‘Real’ sales that is, adjusted for inflation are A B C D 10

approximately constant and keeping up with inflation growing steadily and not keeping up with inflation growing steadily and keeping ahead of inflation falling steadily and not keeping up with inflation

A product has a constant (flat) trend in its sales, and is subject to quarterly seasonal variations as follows: Quarter Seasonality

Q1

Q2

Q3

Q4

50%

50%

50%

50%

Sales last quarter, Q2, were 240 units. Assuming a multiplicative model for the time series, predicted unit sales for the next quarter, Q3, will be closest to A B C D

60 80 120 160


88 90 120 200 498 so A

2

498

90 120 200 28 438 936

so C 3

8th quarter total 8th quarter average actual 2nd quarter variation so D

936 117 120 3

100

Exam Practice Kit: Business Mathematics

4 Average Q1

Seasonal variations Q2

2001 3 2002 51 7 2003 61 Total 112 4 Mean 56 2 Mean 11.5 Adjusted mean 53 1 Best estimate trend seasonal variation 120 70 50

Q3

Q4

99 100

57 49

199 99.5

106 52

102

50

so D 5

x 16 345.12 1.35(16) 323.52 seasonally adjusted 23.62 so 300 so B

6

y 7.112 3.949x seasonal variation 1.12 trend for month 19 y 7.112 (3.949 19) 1.12 y 92 so C

7

The influence of booms and slumps in an industry is a measure of cyclical variations so B

8

Xi

n

i1

so C 9

Quarter real sales 1 2 3 4

109 100 120 110 132 121 145 133

100 109.0 100 109.1 100 109.1 100 109.0

The real series is approximately constant and keeping up with inflation. so A

Time Series 10 Multiplicative model forecast T S Sales last quarter 240 (Q2) Seasonality for Q2 50% S 150 240 Trend 100 160 for Q3 150 S 50 Seasonality 50% 50 Forecast 160 80 100 so B

101

Mock Examination


16 1/2 is equal to A B C D

2

1,600 2,000 2,400 2,800

The numerical value of the expression A B C D

4

√

2DC PR if Q 200, C 10, P 6, R 0.2 then D is equal to In the formula Q

A B C D 3

2 4 8 15.5

(x3)3 when x is equal to 7 is x7

7 17 49 343

If 2x 3y 42 and 5x y 20 then the values of x and y are A B C D

x6 x 10 x5 x3

y 10 y6 y3 y5 102

17

Mock Examination 103 5 If 6x2 15x 25 0 then the value of x is equal to A B C D

6 15 25 cannot be found

6 If b2 4ac is positive there are A B C D

no solutions 1 solution 2 solutions 3 solutions

7 If stock is purchased for £200, what should be selling price be to achieve a profit of 40%? A B C D

£250 £300 £325 £333.33

8 Mr Bachman, Mr Turner and Mr Overdrive are in partnership and have agreed to share profits in the ratio 7:6:5. How much would Mr Bachman expect to receive if the partnership made £180,000? A B C D

£60,000 £70,000 £80,000 £90,000

9 x% of 300 equals x 300 x B 3 3 C x D 3x A

10 An item priced at £105 including VAT of 17 1/2% is reduced in a sale by 25%. The new price before VAT is added is A B C D

£67.02 £69.99 £79.54 £84.56

11 The compound interest on £1,000 at 8% over 10 years is equal to A B C D

£1,159 £1,250 £1,361 £1,452

104


12 How much needs to be invested today at 6% per annum to provide an annuity of £10,000 per annum for ten years starting in five years’ time? A B C D

£25,000 £26,750 £29,150 £30,250

13 How much needs to be invested now at 10% to yield an annual income of £6,000 in perpetuity? A B C D

£50,000 £60,000 £70,000 £80,000

14 If interest rates are 12% which of the following is worth most at present values? A B C D

£100 today £120 in a year’s time £130 in 2 years’ time £140 in 3 years’ time

15 The present value of a stream of 5 annual rental incomes of £10,000 and interest rates are 10%, first one due now is closest to A B C D

£50,000 £45,000 £41,700 £38,200

16 Which sampling technique considers probability proportioned to size? A B C D

random sampling systematic sampling stratified sampling quota sampling

17 If the United Kingdom exports 10% of its invisible earnings to the United States how many degrees would this be represented by in a PIE chart? A B C D

10 degrees 36 degrees 72 degrees 90 degrees

18 In a histogram in which one class interval is two and a half times as wide as the remaining classes, the height to be plotted in relation to the frequency for that class is A B C D

20% 40% 80% 100%

Mock Examination 105 19 The arithmetic mean of the following 10 exam scores was 53 53, 54, 52, x, 47, 48, 49, x, 51, x The value of x was A B C D

54.24 55.61 58.67 59.24

20 Which of the following is a disadvantage of the mode? A B C D

It is difficult to understand. It is affected by extreme values. It cannot be a value of an actual item in the distribution. It is not suitable for mathematical statistics.

21 The standard deviation of 3, 6, 9, 12 and 15 is A B C D

3.59 4.24 5.61 6.23

22 A frequency distribution has a mean of 20, a median of 18 and a standard deviation of 3. Pearson’s coefficient of skewness is A B C D

1 2 3 4

23 Which of the following statements is incorrect? A In a symmetrical distribution, the mean, median and mode will all have the same value. B In a skewed distribution, the median lies between the mean and the mode. C The mode is the number which appears the most often. D In a positively skewed distribution the mode is greater than the mean. 24 The terms of trade is a price index of export prices over import prices and is found by the formula n1 exports 100 n1 imports If the base year 2005 is 100 and the n1 exports in 2006 is 109.5 and the n1 of imports is 106.4, the terms of trade for 2006 were A B C D

102.9 103.1 98.6 96.9

25 Which of the following is an advantage of the Paasche index over the Laspeyre index? A B C D

It is based on the current pattern of consumption. It is easier to calculate since the numerator and denominator don’t change. If does not underestimate inflation. It is comparable with all previous years.


106

26 When an unbiased die is thrown each of the numbers 1–6 has an equal chance of falling uppermost. What is the probability of two sixes being thrown in succession? A B C D

1 in 6 1 in 12 1 in 18 1 in 36

27 If a coin is tossed four times, what is the probability of obtaining at least one head? A B C D

1/16 15/16 1/36 1/48

28 The probability of the home team winning is 0.6, the probability of the away team winning is 0.3 and the probability of a draw is 0.1. What is the probability of the home team not winning. A B C D

0.2 0.3 0.4 0.5

29 A motorist parks his car illegally five times per week. He will be fined £50 if he is caught and will save £10 if he gets away with it. He has a 40% chance of being caught. What is his expected cost adopting this strategy over a week? A B C D

£50 £60 £66 £70

30 A student has a 75% chance of passing Paper 1, 60% chance of passing Paper 2 and a 50% chance of passing Paper 3. What are the chances of this student failing all three papers? A B C D

5% 10% 20% 25%

Questions 31–33 are based on the following information A group of students had a mean mark in an examination of 50% and a standard deviation of 15. 31 What is the probability of a student scoring more than 60%? A B C D

25% 30% 35% 40%

Mock Examination 107 32 The probability of a student scoring between 45 and 55 is approximately? A B C D

20% 26% 32% 48%

33 What is the probability of a student scoring between 55 and 60? A B C D

10% 12% 14% 18%

34 A sample of 100 items from a production line has a mean weight of 80 g with a standard deviation of 4 g. The 95% confidence interval for the mean weight of all items on the production line is between A B C D

79.216 and 80.784 78.356 and 82.424 77.351 and 83.451 75.125 and 84.875

35 Fast Foods Ltd have identified a relationship between net profit x and the number of outlets y x 0.35y 2.2 where x net profit in £millions. If the company plans to have 20 outlets next year, their profit forecast is A B C D

£3.3m £4.4m £4.8m £6.6m

36 If v 0.35, how much of the variation in the dependent variable is explained by the variation of the independent variable? A B C D

35% 12.25% 7.25% 5.05%

37 In a scatter diagram, time is always shown as A B C D

the x axis since it is the dependent variable the x axis since it is an independent variable the y axis since it is the dependent variable the y axis since it is an independent variable

38 If a shopkeeper’s daily sales figures were plotted as a time series, the fact that Saturday’s sales were higher than the other days is an example of A B C D

long-term trend cyclical variation seasonal variation random variation


108

39 Using an additive model, if the sales for the particular day were 100 and the seasonal variation was 5, then the deseasonalised figure is A B C D

95 100 105 impossible to calculate

40 Using a multiplicative model, if the sales for a particular day were 180 and the seasonal variation was 10% then the deseasonalised figure is A B C D

162 170 190 200


161/2 √16 4 so B

2 200

√2D 10 √6 0.2

2D 10 1.2 40,000 1.2 D 20 2,400

40,000

so C 3

The numerical value of the expression x9 x9 x7 x2 49 x7 so C

4

2x 3y 42 (1) 5x y 20 (2) 2x 3y 42 (1) 15x 3y 60 (3) 17x 102 x6 2x 3y 42 12 3y 42 3y 30 y 10 So x 6y 10 so A

(x3)3 where x 7 is x7

Mock Examination 109 b √b2 4ac 2a (15) √(15)2 4 6 25 26 15 √225 600 12 Therefore no real solution since b2 4ac is negative. so D

5 x

6 If b2 4ac is positive there are 2 solutions. so C 7 If profit needs to be 40% of selling price then unit cost must be 60% of selling price. 60% of SP £200 so 100% of SP £333.33 so D 8 Mr Bachman would receive

7 of £180,000 £70,000 18

so B 9 Suppose x% was 10% then x% of 300 would be 3 10% so 3x so D 10 Price before sales tax £105

1 0.75 1.175

£67.02 so A 11

Using v x(1 r)n where x £1,000 r 8% n 10 then v £1,000 (1 0.08)10 £1,159 so A

12 See annuity table – check values of 6% at year 14 and year 4 PV £5,000 (year 14 year 4) £5,000 (9.295 3.465) £29,150 so C 13 Present value at perpetuity is £6,000 so B 14 a £100 b £120 0.893 £107.16 c £130 7.97 £103.61 d £140 0.712 £99.68 so B

1 £60,000 0.1

110


15 £10,000 c (1 3.170) £41,700 so C 16 The sampling technique which considers probability proportional to size is known as stratified sampling. so C 17 If the United Kingdom exports 10% of its invisible earnings to the United States this will account for 36% since there are 360° in a circle. so B 18 If the class interval is 2 1/2 times as wide this represents 5/2 so the height needs to be 2/5 or 40%. so B 19 53 54 52 47 48 49 51 3x 530 354 3x 530 3x 176 x 58.67 so C 20 The mode is not suitable for mathematical statistics. so D 21

x 3 6 9 12 15 x 45

x2 9 36 81 144 225 2 x 495

√495 (45)2 (5) √5 √99 81 √18

4.24 so B 22

3 (20 18) 2 3 so B

23 In a positively skewed distribution the mode is less than the mean so incorrect statement is D. so D 24 Terms of trade

n1 exports 100 n1 imports

109.5 100 102.9 106.4 so A

Mock Examination 111 25 The main advantage that the Paasche index has over the Laspeyre index is that it is based on the current pattern of consumption. so A 26 The probability of two sixes being thrown in succession is 1 1 1 6 6 36 so D 27 Probability of no heads 1 1 1 1 1 2 2 2 2 16 1 15 so at least 1 1 16 16 so B 28 Probability of the home team not winning the away team winning draw 0.3 0.1 0.4 so C 29

40% expected cost

60% expected saving

Day 1 Day 2 Day 3 Day 4 Day 5

20 6 20 6 20 6 20 6 20 6 100 30 (30) 70 Would be better to park legally since cost is £74 and he is only saving £50. so D 30 Chances of failing all three papers 1 2 1 5% 4 5 2 so A 60 50 0.667 15 from normal table area 0.2486 so probability of scoring over 60 0.5000 0.2486 0.2514 so A

31 Z

32

45 50 0.33 0.1293 15 55 50 0.1293 0.33 15 0.2586 so B

112 33


60 50 0.667 0.2486 15 0.1293 55 50 0.33 15 0.1193 so B 4 0.4 grams √100 The 95% confidence limits are 80 1.96 0.4 80 0.784 This means that there is a 95% probability that the mean of all weights is between 79.216 and 80.784. so A

34 Standard error

35 x 0.35 20 2.2 x 7 2.2 x 4.8 so C 36 If v 0.35, then r2 12.25 so 12.25% of the variation is explained by variations in the independent variable. so B 37 In a scatter diagram, time is always shown as the x axis since it is an independent variable. so B 38 The fact that Saturday’s sales were higher than the other days is an example of cyclical variation. so B 39 If the sales for the particular day were 100 and the seasonal variation was 5, then, using an additive model, the deseasonalised figure is 95. so A 40 10% of 200 180 so D

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