Statistics: Problems and Solutions

238 Pages·1973·15.07 MB·English

by  J. Murdoch BSc, ARTC, AMIProdE, J. A. Barnes BSc, ARCS (auth.)

STATISTICS:
PROBLEMS AND SOLUTIONS
A Complete Course in Statistics
by
J. Murdoch BSc, ARTC, AMIProdE
and
J. A. Barnes BSc, ARCS
STATISTICS, PROBLEMS AND SOLUTIONS
BASIC STATISTICS, LABORATORY INSTRUCTION MANUAL
STATISTICAL TABLES FOR SCIENCE ENGINEERING
BUSINESS STUDIES AND MANAGEMENT
STATISTICS:
PROBLEMS AND SOLUTIONS
J. Murdoch, BSc, ARTC, AMIProdE
Head of Statistics and Operational Research Section,
Cranfield Institute of Technology
and
J. A. Barnes, BSc, ARCS
Lecturer in Statistics and Operational Research,
Cranfield Institute of Technology
Palgrave Macmillan
© J. Murdoch and J. A. Barnes 1973
All rights reserved. No part of this publication may be reproduced or transmitted,
in any form or by any means, without permission
First published 1973
Published by
THE MACMILLAN PRESS LTD
London and Basingstoke
Associated companies in New York Toronto
Melbourne Dublin Johannesburg and Madras
SBN 333 12017 5
ISBN 978-0-333-12017-0 ISBN 978-1-349-01063-9 (eBook)
DOI 10.1007/978-1-349-01063-9
Preface
Statistics is often regarded as a boring, and therefore difficult, subject
particularly by those whose previous experience has not produced any real
need to understand variation and to make appropriate allowances for it. The
subject can certainly be presented in a boring way and in much advanced work
can be conceptually and mathematically very difficult indeed.
However for most people a simple but informed approach to the collection,
analysis and interpretation of numerical information is of tremendous benefit
to them in reducing some of the uncertainties involved in decision making. It is
a pity that many formal courses of statistics appear to frighten people away
from achieving this basic attitude usually through failing to relate the theory to
practical applications.
This book, whose chapters each contain a brief summary of the main
concepts and methods, is intended to show, through worked examples, some of
the practical applications of simple statistical methods and so to stimulate
interest. In order to establish firmly the basic concepts, a more detailed
treatment of the theory is given in chapters 1 and 2. Some examples of a more
academic nature are also given to illustrate the way of thinking about problems.
Each chapter contains problems for the reader to attempt, the solutions to
these being discussed in some detail, particularly in relation to the inferences
that can validly be drawn even in those cases where the numbers have been put
into the correct 'textbook formula' for the situation.
This book will not only greatly assist students to gain a better appreciation
of the basic concepts and use of the theory, but will also be of interest to
personnel in industry and commerce, enabling them to see the range of
applic:ltion of basic statistical concepts.
For the application of basic statistics, it is essential that statistical tables
are used to reduce the computation to a minimum. The tables used are those
by the authors, Statistical Tables, a companion volume in this series of
publications on statistics. The third book, Basic Statistics: Laboratory
Instruction Manual, designed to be used with the Cranfield Statistical Teaching
Aids is referred to here and, in addition, some experiments are suggested for
v
vi Preface
the reader to perform to help him understand the concepts involved. In the
chapters of this book references to Statistical Tables for Science, Engineering
and Management are followed by an asterisk to distinguish them from
references to tables in this book.
The problems and examples given represent work by the authors over many
years and every attempt has been made to-select a representative range to
illustrate the basic concepts and application of the techniques. The authors
would like to apologise if inadvertently examples which they have used have
previously been published. It is extremely difficult in collating a problem book
such as this to avoid some cases of duplication.
It is hoped that this new book, together with its two companion books, will
form the basis of an effective approach to the teaching of statistics, and
certainly the results from its trials at Cranfield have proved very stimulating.
J. Murdoch
Cranfield J. A. Barnes
Contents
list of symbols xi
Probability theory
1.2.1 Introduction 1
1.2.2 Measurement of probability
1.2.3 Experimental Measurement of Probab~lity 2
1.2.4 Basic laws of probability 2
1.2.5 Conditional probability 6
1.2.6 Theory of groups 10
1.2.7 Mathematical expectation 11
1.2.8 Geometric probability 12
1.2.9 Introduction to the hypergeometric law 13
1.2.10 Introduction to the binomial law 14
1.2.11 Management decision theory 15
1.3 Problems 17
1.4 Worked solutions 19
1.5 Practical experiments 25
Appendix I-specimen experimental results 27
2 Theory of distributions 32
2.2.1 Introduction 32
2.2.2 Frequency distributions 33
2.2.3 Probability distributions 35
2.2.4 Populations 35
2.2.5 Moments of distribution 37
2.2.6 Summary of terms 38
2.2.7 Types of distribution 40
2.2.8 Computation of moments 42
2.2.9 Sheppard's correction 45
2.3 Problems 45
2.4 Worked solutions 48
vii
viii Contents
2.5 Practical experiments 60
2.5.1 The drinking straw experiment 60
2.5.2 The shove halfpenny experiment 61
2.5.3 The Quincunx 61
3 Hypergeometric binomial and Poisson distributions 63
3.2.1 Hypergeometric law 63
3.2.2 Binomial law 63
3.2.3 Poisson law 63
3.2.4 Examples of the use of the distributions 65
3.2.5 Examples of the Poisson distribution 68
3.3 Problems 72
3.4 Worked solutions 73
3.5 Practical experiments 76
Appendix I-binomial experiment with specimen results 77
4 Normal distribution 80
4.2.1 Introduction 80
4.2.2 Equation of normal curve 80
4.2.3 Standardised variate 81
4.2.4 Area under normal curve 81
4.2.5 Percentage points of the normal distribution 82
4.2.6 Ordinates of the normal curve 82
4.2.7 Fitting a normal distribution to data 82
4.2.8 Arithmetic probability paper 82
4.2.9 Worked examples 83
4.3 Problems 89
4.4 Worked solutions 92
4.5 Practical experiments 102
Appendix I-Experiment 10 of Laboratory Manual* 102
Appendix 2-Experiment 11 of Laboratory Manual* 105
5 Relationship between the basic distributions 109
5.2 Resume of theory 109
5.2.1 Hypergeometric, binomial and Poisson approximations 111
5.2.2. Normal approximation to Poisson 112
5.2.3 Examples of use of approximations 113
5.2.4 Examples of special interest 115
5.3 Problems 118
5.4 Worked solution~ 119
Appendix I-Experiment 8 of Laboratory Manual* 122
Contents ix
6 Distribution of linear functions of variables 124
6.2.1 Linear combination of variates 124
6.2.2 Sum of n variates 127
6.2.3 Distribution of sample mean 127
6.2.4 Central limit theorem 128
6.2.5 Sum of two means 129
6.3 Problems 130
6.4 Worked solutions 133
Appendix I-Experiment 12 of Laboratory Manual* 141
7 Estimation and significance testing (I)-'large sample' methods 145
7.2.1 Point estimators 145
7.2.2 Confidence intervals 145
7.2.3 Hypothesis testing 146
7.2.4 Errors involved in hypothesis testing 146
7.2.5 Hypothesis (significance) testing 147
7.2.6 Sample size 147
7.2.7 Tests for means and proportions 147
7.2.8 Practical significance 149
7.2.9 Exact and approximate tests 149
7.2.10 Interpretation of significant results 150
7.2.11 Worked examples 150
7.3 Problems 162
7.4 Worked solutions 163
8 Sampling theory and significance testing (11)-'t', 'F' and X2 tests 170
8.2.1 Unbiased estimate of population variance 170
8.2.2 Degrees of freedom 171
8.2.3 The 'u '-test with small samples 171
8.2.4 The 't'-test of significance 172
8.2.5 The 'F'-test of significance 174
8.2.6 The 'X2 '-test of significance 175
8.2.7 One-and two-tailed tests 177
8.2.8 Worked examples 177
8.3 Problems 182
8.4 Worked solutions 184
8.5 Practical experiments 192
Appendix I-Experiment 14 of Laboratory Manual* 193
9 Linear regression theory 197
9.2.1 Basic concepts 197
9.2.2 Assumptions 198