HEINEMANN SENIOR MATHEMATICS

HEINEMANN SENIOR MATHEMATICS

_)! __

tieinemann Educational Australia

i

,, I

'

r

i

i·

I \ ,,

,, ·,,

,·

I

I

i

(

l

Heinemann Educational Australia

a division of the Octopus Publishing Group Australia Pty Ltd 22 Salmon Street, Port Melbourne, Victoria 3207

Offices in Sydney, Brisbane and Adelaide. Associated companies, branches and representatives throughout the world.

©J.B. Fitzpatrick and P. L. Galbraith 1990 First published 1990

Reprinted 1991

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form by any means whatsoever without the prior permission of the c_opyright owner. Apply in writing to the publishers.

Edited by Scharlaine Cairns, Charlie C. Editorial Pty Ltd Designed by Tom Kurema

Illustrations by Gavin Mount

Keying and preparation of disks by Tricia Randle

Typeset in Times Roman by Savage Type Pty Ltd, Brisbane Printed in Singapore by Chong Moh Offset Printing

National Library of Australia Cataloguing-in-publication data:

Fitzpatrick, J. B. (John Bernard). Reasoning and data

Includes index. ISBN O 85859 527 3.

l . Mathematics. I. Galbraith, P. (Peter). II. Henry, Bruce. Ill. Title. (Series: Heinemann senior mathematics).

Contents

(Projects and Investigations are identified by

[;J

and

U

respectively.)

Acknowledgements (x)

Preface [xi)

Chapter 1 Statistics 1

1.1

Graphical representation of data 2

(;I

1.2

Limited-over cricket 9

1.3

Continuous and discrete data 10

1.4

Frequency distribution 10

1.5

Histograms 12

1.6

Frequency polygons 14

1. 7

Measures of central tendency 17

1.8

Measurement of dispersion 22

(;I

1.9

Multi-lingual 'Scrabble' 30

Chapter 2 Probability 31

2.1

Complementary events 35

2.2

Life tables (1984) 36

2.3

Finite sample space 40

2.4

Mutually exclusive events 41

2.5

Successive outcomes 45

2.6

Independent events 50

2.7

Conditional probability: a reduced sample space 61

2.8

Baye·s• Theorem 70

Chapter 3 Permutations and combinations 77

3.1

Permutations 78

3.2

The multiplication principle 79

3.3

Mutually exclusive operations: addition principle 81

3.4

Definition of permutation 81

3.5

The symbol np

r

83

3;6

Arrangements with restriction� 85

3. 7

Arrangements in a circle 88

3.8

Number of arrangements of

n

objects

in a row, when they are not all different 89

(v)

' ·

/

3.9

Combinations 92

3.10

_{The symbol (;) or} ncr 93

3.11

Probability associated with permutations and combinations 100

Chapter 4 Binomial distribution

111

4.1

Binomial theorem

112-4.2

Binomial probability distribution 117

4.3

Mean and v�riance of a discrete random variable 127

4.4

Mean and variance of a binomial distribution 129

l];I

4.5

Coin tossing 133

l];I

4.6

Computer simulation of binomial experiments 134

Chapter 5 Other discrete probability distributions

137

5.1

� Hypergeometric distribution: sampling without replacement 138

5.2

Mean and variance of a hypergeometric distribution 143

5.3

Geometric distribution 146

5.4

Poisson distribution 149

5.5

Exponential distribution 157

l];I

5.6

Poisson distribution project 157

5.7

Probability and matrices: Markov chains 159

Chapter 6 Normal distribution

167

6.1

The normal distribution 168

6.2

Standard normal curve 169

6.3

Normal approximation to binomial distribution 179

6.4

Probability limits for a single value of the normal variable 182

6.5

Probability limits for the sample mean of n values of the variable 186

6.6

Confidence limits 188

Revision exercises (Chapters 1 to 6)

197

Chapter 7 Problem-solving and investigations 20s

7.1

Problems 206

D

7 .2

Investigations 214

Chapter 8 Related variables

219

8.1

Scatter diagrams 220

8.2

Regression of

yon x

221

8.3

Method of least squares 222

8.4

Bivariate distributions: two regression lines 224

8.5

Correlation 231

8.6

Correlation and causation 237

D

8. 7

Correlation investigation 239

8.8

Non-linear relationships 240

8.10

Analysis of a time series 248

8.11

Measures of trend 248

8.12

Measurement of seasonal variations 253

8.13

Forecasting using single moving average 256

Chapter 9 Non-parametric statistical tests

9.1

Hypothesis testing: stating the hypotheses 264

9.2

The sign test 265

9.3

Significance level 266

9.4

The steps in performing a statistical test 266

9.5

Binomial test of percentiles 267

9.6

Wilcoxon test for two independent samples 270

9. 7

Dealing with ties 272

9.8

Dealing with large samples 272

9.9

Permutation test 273

9.10

Statistical project 277

9.11

; . The Chi-square test 277

9.12

Degrees of freedom, v 279

9.13

x2 test for a Poisson distribution 281

9.14

x2 _{test for a normal distribution 282}

9.15

x2 _{test for a binomial distribution 284}

9.16

Contingency tables 287 (;J

9.17

Newspaper poll 293

9.18

Die-tossing program 293

9.19

Tables 295

Chapter 10 Graphs arid optimisation

297

10.1

Graph theory 298

10.2

Basic definitions and properties 299

10.3

The handshaking lemma 301

10.4

Isomorphic graphs 302

10.5

Cycles and trees 307

10.6

Applications to network problems 308

10. 7

Planar graphs 310

10.8

Eulerian paths 313

10.9

Fleury's Algorithm 316

10.10

Network inspection problems 317

10.11

Shortest path problems 318

10.12

Hamiltonian graphs 320

10.13

The travelling sales representative problem 320 (;J

10.14

Road network 326

D

10.15

Chemical molecules 327

10.16

Digraphs (directed graphs) 328

10.17

Matrix representation 328

10.18-.

:Applications of digraphs 331 (;J

10.19

Graphing projects 339

/

Chapter 11 Logic and reasoning

341

11.1

Propositions 342

11.2

Negation, -p 342

11.3

Set notation 343

11.4

Conjunctionp /\ q 344

11.5

Disjunctionp v q 345

11.6

Conditional statements p � q 349

11. 7

Converse, inverse and contrapositive 350

11.8

Equivalencep +-+ q 351

11.9

Tautologies 358

11.10

Negation of compound sentences 361

11.11

Validity of arguments 363

11.12

Use of tautologies 366

11.13

Quantifiers 368

Chapter 12 Methods of proof

373

12.1

Mathematical proof 374

12.2

Necessary and sufficient conditions 375

12.3

Proof patterns in mathematics 375

12.4

Indirect proof 378

12.5

Proof by counter-example 379

12.6

Famous proofs from antiquity 381

12. 7

Mathematical induction 384

12.8

Problem solving and investigations 390

D

12.9

Logic investigations 392

(;I

12.10

Logic projects 394

12.11

Finite differences 395

D

12.12

Cheese slicing 398

D

12.13

Pizza party 398

D

12.14

The twelve days of Christmas 398

(;I

12.15

Number patterns 399

Chapter 13 Boolean algebra

401

13.1

Laws of set algebra 403

13.2

Boolean algebra 404

13.3

Principle of Duality 405

13.4

Theorems in Boolean algebra 405

13.5

De Morgan's laws 407

D

13.6

Boolean algebra investigation 409

13.7

Examples of Boolean algebras 410

13.8

Electrical circuits 412

13.9

Simplification of circuits 413

13.10

Boolean functions 415

13.11

Disjunctive form 415

13.12

Conjunctive form 417

Chapter 14 Calculus [extension)

423

14.1 Power series for � 424

14.2 Antidifferentiation by parts 427 14.3 Other density functions 429

14.4 Measures of location for probability distributions 438 14.5 The mean (expected value) of g(X) 444

14.6 Variance and standard deviation 444

Chapter 15 Euclidean geometry (extension)

449

15.1 Assumptions 450

15.2 Angle properties of atriangle 451 15.3 Congruent triangles 456

15.4 Similar triangles 464

15.5 Theorem of Pythagoras 468

15.6 Circle theorems 473 15.7 Cyclic quadrilaterals 478 15.8 Tangents to a circle 482 15.9 Alternate segment 484

15.10 Intersecting chords of a circle 488

15.11 Concurrency theorems 490

Summary

495

Answers

503

Index

531

Acknowledgements

The authors with to express their thanks to Mr Ted Byrt, formerly of State College Rusden Campus, for his contribution and helpful suggestions in the area of statistics.

The authors and publisher would like to thank the following individuals and organisations for their assistance in providing photographs and for their permission to reproduce copyright material:

Charles Ciurleo, pp. 77 (a, b, c), 137 (b) and 219; D. A. Heffernan, p. 401; The Herald

& Weekly Times Ltd, Melbourne, pp. 77 (d), 263 and 423; Tattersall Sweep Consultation, pp. 205; Tubemakers of Australia Ltd, p. 167.

Preface

Reasoning and Data provides a comprehensive coverage of the compulsory sections of the

unit, together with detailed coverage of eight of the content clusters. The book also provides for study of Reasoning and Data at the extension level, with coverage of the probability, statistics, and algebra requirements together with two selections (calculus and geometry) from the additional study areas.

With respect to the work requirements, essential content in the area of probability is contained within Chapters 2, 4, 5 and 6. The compulsory statistics material is contained in Chapters 1, 4, 5 and 6. Chapter 1 is an introduction, consolidating aspects of data

representation that will have been studied to varying degrees in past years. The other chapters systematically introduce discrete and continuous distributions together with their special features, and related calculations of statistical measures and estimates of parameters.

The logic requirements are provided for within Chapters 2, 10, 12 and 13. Set diagrams are

utilised in probability work (Chapter 2) and also in the chapters on logic and reasoning (Chapter 11) and Boolean algebra (Chapter 13). The concept and application of proof appears in the chapters on logic and reasoning (Chapter 11), graphs and optimisation (Chapter 10), methods of proof (Chapter 12) and Boolean algebra (Chapter 13). 'Graphs and optimisation' (Chapter 10) contains all the material necessary for the study of undirected graphs.

The algebra section is well covered. Chapter 3 contains applications of combinations; basic equation solving and formula manipulation is required regularly throughout almost all chapters; set algebra is used widely in Chapters 3, 11 and 13; sequences and series are applied in Chapters 8 and 12, and Chapter 8 also includes work on non-linear relationships. The companion volumes Space and Number and Change and Approximation contain additional material that systematically addresses analytical and numerical methods for solving equations and inequations.

Clusters of content

The following chapters contain material that enables comprehensive coverage of the nominated clusters.

Combinations Chapters 3 and 4

Sampling processes Chapter 4

Probability distributions -geometric, Poisson and exponential Chapter 5

Time series analysis and economic statistics Chapter 8

Correlation and regression Chapter 8

Non-parametric statistics Chapter 9

Logic and proof Chapters 11 and 12

Boolean Algebra Chapter 13

In addition, substantial amounts of material pertaining to the Clusters (Random sampling, Estimation and confidence intervals, and Directed graphs) are also included.

/ /

For the extension course, Chapters 2, 3, 4 and 6 contain extension material for probability;

Chapter 6 provides extension material for statistics; and Chapter 3 provides extension

material for algebra.

Within the additional area of study, two options are provided; 'Calculus extension' (Chapter 14) and 'Euclidean geometry extension' (Chapter 15).

The treatment of the subject matter emphasises coherence so that, where relevant, extension material appears as a natural development bf the core material. Chapter 7 'Problem solving and investigations' provides material particularly geared to problem solving, modelling, and project work.

Features of the presentation include:

• a systematic and thorough introduction to, and consolidation of, content material to promote concept understanding and facility in skills and standard applications. Numerous worked examples and sets of exercises are included to this end, including sets of revision exercises.

• provision of problem-solving examples, modelling situations, investigations and project material integrated through the chapters, in addition to those provided in Chapter 7. • integration of the electronic calculator throughout, and provision of computer-based

learning tasks for concept learning, applications and investigation and project work. Project material is defined in terms of its nature rather than its length. School projects of varying lengths may be obtained by combining one or more text-based projects. Text-based Projects and Investigations are frequently presented in a sequential fashion so that

variations between students can be provided for, e.g. not every student may be required to complete every part of such an activity. A computer application often forms the final section of a Project I Investigation and can be retained or omitted without otherwise affecting the structure.

The authors endorse the spirit and intent of the general course structure and its work

requirements. It is expected that many effective modelling s_ituatfons, investigations and

projects will be designed with the local school environment in mind. This book provides a supporting base upon which such local emphasis can be built, while at the same time containing more than sufficient material to meet the work requirements in all areas.

Computer Policy Statement

Throughout this text and its companion volumes there are a number of short programs for carrying out specific mathematical tasks. Students are also given the opportunity to write their own programs to help in some of the exercises, applications and models.

It is not the place in a text such as this to teach the elements of computing. These will have

been mastered already by anyone wishing to use a computer productively with this book. We are aware that there is a degree of debate about programming languages such as BASIC, LOGO and PASCAL, and each has its supporters. We have chosen to use the BASIC language, not because it is the best, but because it is the most universally available on the facilities available to most students. Those who wish to work in another language have the opportunity to do so, by converting the coding that is provided or by working directly from the verbal context of the exercises, applications and models.