Numbers, Hypotheses and Conclusions: A Course in Statistics for the Social Sciences

542 Pages·2004·13.31 MB·English

by  Colin Tredoux

Numbers, Hypotheses
& Conclusions
A course in Statistics for the
Social Sciences
Editors:
Colin Tredoux (University of Cape Town)
Kevin Durrheim (University of Natal)
UCT
PRESS
Numbers, Hypotheses & Conclusions: ACourse in Statistics for the Social Sciences
(c) UCT Press, 2002
PO Box 24309
Lansdowne 7779
First published 2002
ISBN 1-919713-31-X
This book is copyright under the Berne Convention. In terms of the Copyright Act,
No. 98 of 1978, no part of this book may be reproduced or transmitted in any form
or by any means, electronic or mechanical, including photocopying, recording or
by any information storage and retrieval system, without permission in writing
from the Publisher.
Copy editing: Ann Kantey, Cape Town
Proofreading: Alex Potter of FPPProductions, Hermanus
DTP: RHT desktop publishing cc, Durbanville
Cover design: Eugene Badenhorst of The Pumphaus Design Studio cc, Cape Town
Indexing: Jan Schaafsma
Printing and binding:
Set in Palintino and Rotis Semi Sans
The authors and publisher have made every effort to obtain permission for and
acknowledge the use of copyright material. Should any infringement of copyright
have occurred, please contact the publisher and every effort will be made to rectify
omissions or errors, in the event of a reprint or new edition.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
SECTION 1: Statistics
Tutorial 1. Numbers, variables, and measures . . . . . . . . . . . . 2
Tutorial 2. Displaying data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Tutorial 3. Central tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Tutorial 4. Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Tutorial 5. Probability and theoretical distributions . . . . . . . . 70
Tutorial 6. The standard normal distribution . . . . . . . . . . . . . 90
Tutorial 7. The sampling distribution of the mean . . . . . . . . . 108
Tutorial 8. Hypothesis testing: the z-test . . . . . . . . . . . . . . . . . 127
Tutorial 9. T-tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Tutorial 10. Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Tutorial 11. Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Tutorial 12. Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Tutorial 13. Statistical power . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Tutorial 14. The logic of analysis of variance . . . . . . . . . . . . . . 252
Tutorial 15. One-way analysis of variance . . . . . . . . . . . . . . . . 271
Tutorial 16. Factorial analysis of variance . . . . . . . . . . . . . . . . . 289
Tutorial 17. Repeated measures analysis of variance . . . . . . . . 311
Tutorial 18. Multiple regression . . . . . . . . . . . . . . . . . . . . . . . . . 338
Tutorial 19. Chi-squaretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Tutorial 20. Distribution-free tests . . . . . . . . . . . . . . . . . . . . . . . 385
Tutorial 21. Meta analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Tutorial 22. Statistical reasoning . . . . . . . . . . . . . . . . . . . . . . . . 423
SECTION 2: Mathematics and Software Support
Tutorial 23. Basic work with numbers . . . . . . . . . . . . . . . . . . . . 444
Tutorial 24. Equations, substitution, and summation . . . . . . . 461
Tutorial 25. Reading and understanding graphs . . . . . . . . . . . 472
Appendix 1: Statistical tables . . . . . . . . . . . . . . . . . . . . . . . . . . 485
z-table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
t-table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
F ((cid:2)(cid:3) 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
F ((cid:2)(cid:3) 0.01) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Tukey’s Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Chi-square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
Wilcoxon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
Mann-Witney ((cid:2)(cid:3)0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
Mann-Witney ((cid:2)(cid:3)0.025) . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Appendix 2: Starting with SPSS® . . . . . . . . . . . . . . . . . . . . . . . 496
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Preface
This book is the core text for a tutorial programme in statistics for
social science and humanities students. Although it can be used as
a self-sufficient introduction to statistics, we recommend that it be
used alongside a structured set of exercises, worked examples,
and activities. An excellent collection of support materials is
available on a course web page (http://www.uct.ac.za/depts/
psychology/plato/numbers/) and on the accompanying CD.
The tutorial programme is aimed at a diverse group of disci-
plines. Students of Psychology, Sociology, Social Work, Anthro-
pology, Education, and Political Studies – indeed, all students in the
social sciences - will be able to use this book. Whatever your area of
study, we hope that it will stimulate you, teach you, tease you, and
promote you to the rank of inveterate inquirer. The skills we aim to
impart are central to any modern knowledge-based enterprise.
They are taught all over the world in programmes that are serious
about research, and thus provide a universal language for the social
sciences. More than that, they underpin successful theory and
application in almost every field of enquiry in the social sciences.
Indeed, expertise in quantitative methods is one of the strongest
transferable skills taught in the social sciences, and many employ-
ers demand some level of competence in this area.
Social science students differ in terms of their preparation for
courses in quantitative methods. They come from a diversity of
disciplines and backgrounds. This means that there will also be dif-
ferences in their level of mathematics proficiency. We acknowledge
this, and see it as a challenge. For this reason, we have included a
substantial collection of revision material in the appendices and on
the accompanying CD. All students entering a course in quantita-
tive methods will benefit by revising their school mathematics with
this material.
Our emphasis in the course, as a whole, is on statistical concepts
and techniques. We promote the use of simple mathematical mani-
pulations and calculations to aid understanding. It is important for
you to know how to do some basic statistical calculations, and we
encourage you to improve your skills no matter what your starting
level of proficiency is. In the modern world, however, there are
many aids to the error-prone activity of statistical computation, and
we specifically show you how to use calculators, spreadsheet pro-
grams, and statistical software packages. In the early tutorials, we
emphasise calculator and spreadsheet work, and in later tutorials we
assume the availability of a statistical package. We have provided
V
material showing you how to use SPSS® for particular statistical
analyses, but you could also use a package like STATISTCA, and we
have included extensive support material for that package in the
accompanying CD.
We have tried to enhance your experience in this text by the
extensive preparation of activities, interest boxes, application boxes,
graphic material, exercises, worked solutions to problems, and
Internet links and resources. You should use these to your advant-
age. We suggest that you keep a calculator at hand whenever you
read the text, and complete the activities and exercises. When you
have finished studying the text, we strongly recommend that you
spend some time browsing the CD for additional worked examples,
exercises, and worked solutions. We also encourage you to use the
Internet links, as there are many interactive web sites that demon-
strate statistical concepts and techniques in an interactive manner.
Colin Tredoux and Kevin Durrheim, March 2002
VI
Acknowledgements
We would like to acknowledge the following people for support
and guidance during the long process of writing and editing the
book:
• Diane Gascoigne, Aimée Tredoux, and Aleks Durrheim.
• Solani Ngobeni, Sandie Vahl, Fiona Wakelin and Glenda Younge
from University of Cape Town Press.
• Lance Lachenicht, Martin Terre Blanche, David Nunez, Judy
Austin, Gillian Finchilescu and Ingrid Palmary, all contributors
to this book.
List of Contributors
Judy Austin, Department of Psychology, University of Natal
Kevin Durrheim, Department of Psychology, University of Natal
Gillian Finchilescu, Department of Psychology, University of Cape
Town
Lance Lachenicht, Department of Psychology, University of Cape
Town
David Nunez, Department of Psychology, University of Cape Town
Ingrid Palmary, Department of Psychology, University of Natal
Martin Terre Blanche, Department of Psychology, University of
South Africa
Colin Tredoux, Department of Psychology, University of Cape Town
VII
Glossary of Symbols
(cid:2) Alpha (Type I error rate)
β Beta (Type II error rate)
δ Delta, a parameter used to determine power of a statistical test
µ Mu, the population mean
ρ Rho, the population correlation coefficient
σ Sigma (lower case), the population standard deviation
Σ Sigma (upper case), the arithmetic summation operator
χ2 Chi-square statistic, or Chi-square distribution
η2 Eta-square, a measure of effect size
φ2 Phi square, the mean square contingency coefficient
σ2 Sigma square, the population variance
φ Cramer’s V, a measure of effect size in contingency table analysis
c
a The intercept coefficient in regression analysis
b The slope coefficient in regression analysis
D Difference between two scores
d Effect size
E Expected frequency
F F distribution, or F ratio
k Number of groups in a design
MS Mean Square
N Population size
n Sample size
O Observed frequency
p Probability
Q Tukey’s Q statistic (studentized range statistic)
R Multiple regression coefficient
r The Pearson product moment correlation coefficient
r2 Square of r; coefficient of determination
R2 Square of the multiple regression coefficient; degree of linear model fit
r Spearman’s rank correlation coefficient
s
s2 Sample variance
S2 Standard error of the mean
x
SS Sums of squares
t The tstatistic, used to test hypotheses about mean differences; also the
tprobability distribution
x Sample mean
y’ y prime, the predicted score in a regression equation
z Standard normal deviate
VIII
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Section 1
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• Statistics