© J. Y. Opoku, 2004

ISBN: 9988-0-2352-9Revised EditionFirst Published 2004, New Impression 2005

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any other information storage and retrieval system, without permission in writing from the copyright holder.

Typeset by: Petrine Esi Agbenyega, Cape Coast.Tel: 024-3003967

Produced in Ghana by:

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TOAll my former students of Statistics for Psychologists who’s searching questions during lectures prompted me to write this book.

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ABRIDGED CONTENTS

Preface and AcknowledgementsPreface to the Revised EditionForewordOrganization of the Book

SECTION A: THEORETICAL PERSPECTIVESChapter 1: General Introduction 1.1: Descriptive and Inferential Statistics 1.2: Summation Notation 1.3: Measurement and Levels of Measurement in the Social

Sciences 1.4: The Mean, Standard Deviation, and Variance of

Interval and Ratio Data 1.5: Chapter Summary

Chapter 2:

Introduction to the General Strategy of Inferential Statistics

2.1:

Introduction

2.2:

The Coin Tossing Experiment

2.3:

Inferential Statistics and the Concept of Probability

2.4:

The Probability of Independent Events and the Coin Tossing Experiment

2.5:

The .05 Decision Rule

2.6: The Standard Error of the Mean and the Normal Curve Distribution

2.7: Hypothesis Testing 2.8: Chapter Summary

SECTION B: SOME SELECTED PARAMETRIC STATISTICAL TESTS AND THEIR

APPLICATIONS

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Chapter 3: Inferences about the Mean of a Single Population

3.1 Introduction 3.2:

The Z Test

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3.3:

The t Test based on a Single Population

3.4:

Chapter Summary

3.5:

General Steps to Follow in Answering a Question in Inferential Statistics

3.6:

Worked Example on the Z Test

3.7:

Worked Example on the t Test based on a Single Population

3.8: Exercises on the Z Test and the t Test based on a Single Population

Chapter 4: The t Test for the Difference between the Means of Two Populations (The

Independent t Test) 4.1:

Introduction

4.2:

The Standard Error of the Difference

4.3:

The t Test for the Difference between the Means of 2 Populations

4.4:

Chapter Summary

4.5:

Worked Example on the Independent t Test

4.6:

Exercises on the Independent t Test

Chapter 5: The t Test for Matched or Correlated Samples

5.1:

Introduction

5.2:

The Standard Error of the Mean for Correlated Samples

5.3:

The t Test for Matched (Correlated) Samples

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5.4:

Chapter Summary

5.5:

Worked Example on the Correlated t Test

5.6:

Exercises on the Correlated t Test

5.7:

Addendum: The Correlated t Test and Analysis of Variance

Chapter 6:

One-Way Analysis of Variance (One-Way ANOVA)

6.1: Introduction 6.2: The General Logic of ANOVA 6.3: Computational Procedures 6.4: Post-hoc Analysis: Multiple Comparisons following a

Significant F Test 6.5: Chapter Summary 6.6: Worked Example on the One-Way ANOVA 6.7: Exercises on the One-Way ANOVA 6.8: Addendum: One- and Two-Factor Analysis of

Variance

Chapter 7:

Linear Correlation and Prediction: The Pearson Product- Moment Correlation Coefficient (r)

7.1: Introduction: The Correlation Coefficient 7.2:

The Pearson Product-Moment Correlation Coefficient (r)

7.3: Testing the Significance of the Pearson r 7.4: Worked Example on the Pearson r Test 7.5: Exercises on the Pearson r Test

SECTION C: SOME SELECTED NONPARAMETRIC STATISTICAL TESTS AND THEIR APPLICATIONS

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Chapter 8:

Linear Correlation and Prediction: The Spearman Rank- Order Correlation Coefficient ( )

8.1:

Nonparametric Statistical Tests

8.2:

The Spearman Rank-Order Correlation Coefficient ( )

8.3:

Chapter Summary

8.4:

Worked Example on the Spearman Test

8.5:

Exercises on the Spearman Test

Chapter 9: The Chi Square ( ) Test 9.1: Introduction 9.2: One- Variable Problems with the Chi Square Test 9.3: Hypothesis Testing with the Chi Square Test 9.4: Two-Variable Problems with the Chi Square Test 9.5: Alternative Computation of Chi Square from a

Contingency Table 9.6: Measures of Strength of Association 9.7: Chapter Summary 9.8: Worked Example on the Chi Square Test 9.9 Exercises on the Chi Square Test

Chapter 10: The Mann-Whitney U Test 10.1: Introduction 10.2: Method 10.3: The Mann-Whitney U Test for very Small Samples 10.4: The Mann-Whitney U Test when is between 9 and

20 10.5: Computing the Value of U 10.6: The Case of Large Samples ( ) 10.7: The Problem with Tied Ranks 10.8: Chapter Summary 10.9: Worked Example on the Mann-Whitney U Test

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10.10: Exercises on the Mann-Whitney U Test

Chapter 11: The Wilcoxon Matched-Pairs Signed-Ranks Test

11.1: Introduction 11.2: Rationale and Method 11.3: Method for Small Samples 11.4 An Example for a Small Sample 11.5: Method and an Example for Large Samples 11.6: Chapter Summary 11.7: Worked Example on the Wilcoxon Matched-Pairs

Signed-Ranks Test 11.8: Exercises on the Wilcoxon Matched-Pairs Signed-

Ranks Test

Chapter 12: The Kruskal-Wallis One-Way Analysis of Variance by Ranks Test (H Test)

12.1:

Introduction

12.2:

Method

12.3:

Example for Small Samples

12.4 Example for Large Samples (each ) 12.5:

The Problem with Tied Ranks

12.6:

Multiple Comparisons following a Significant H Test

12.7 Chapter Summary 12.8:

Worked Example on the Kruskal-Wallis H Test

12.9:

Exercises on the Kruskal-Wallis H Test

Important RemindersEpilogue

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Recommended Reading

APPENDICES

Appendix 1: Some Selected Useful Computational Formulae

Appendix 2: Some Selected Statistical Tables

Appendix 3: Guide to the Solution of the Exercises

INDEX

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PREFACE AND ACKNOWLEDGEMENTSA firm grounding in Statistics is a sine qua non for research in the Social Sciences and Education. Despite this truth, very many students shy away from Statistics because they believe it is a difficult course. Many students have the erroneous impression that “Statistics is Mathematics”, and the negative attitude towards Mathematics that they unfortunately develop during their primary and secondary school years is carried to the tertiary level.

As a teacher of “Psychological Statistics” for over two decades, I have tried to change this negative attitude towards Statistics by adopting a completely non-mathematical approach in my teaching. I must confess that I do not think that I have succeeded in my mission mainly because of the large student numbers in Statistics classes that makes it difficult to attend to individual problems. Tutorials are also no longer effective for the same reason.

This book, written in very simple language, is an attempt to offer tutorials in Inferential Social Statistics to students “from a distance” . The book is written with deliberate repetitive emphasis on important basic concepts that some readers with good knowledge in Inferential Statistics may find irritating. However, my decision to stick to this style of writing stems from my aim of writing a book for the reader with little or no knowledge in Inferential Statistics, and the fact that repetition and emphasis aid learning are well known indisputable psychological principles of learning.

The book is based on problems that commonly confront students in a course in Inferential Social Statistics that many textbooks in Statistics do not adequately address. I wish to draw the attention of all users of this book that the book is NOT a textbook in Inferential Social Statistics. The topics treated represent an attempt to clarify basic principles and concepts and applications of these principles and concepts that students find difficult to understand in an Inferential Social Statistics course. My advice is that the book should be used in formal tutorial classes and informal (study group) discussions.

Some readers may be disappointed to find that the book dwells solely on Inferential Statistics. This is deliberate because I believe Descriptive

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Statistics is a course that is easily understood by students. For those who may be weak in Descriptive Statistics, it is absolutely necessary and essential that they refresh their memories in this area in order to gain better understanding of the principles, concepts, and applications explained in the book. The book is written with the aim of covering a semester’s course in Inferential Social Statistics. For this reason, only important selected topics in Parametric and Nonparametric Statistics are treated.

The book is lavishly illustrated with examples and this should make it possible for all users of the book to easily understand the theoretical concepts discussed. Also, unlike many textbooks in Statistics, an attempt is made to relate statistical principles and concepts to empirical research.

It is my hope that all who seriously read this book will find that they have at last found a remedy to the phobia surrounding Inferential Social Statistics they will discover the joy of being “champions” in Inferential Social Statistics.

I wish to thank the following for encouraging and literally “pushing” me to write this book: Professor G.K. Nukunya of the Department of Sociology, University of Ghana, Legon; my numerous former students who continued their education in Europe and the Americas, after receiving tuition from me in Psychological Statistics; and the thousands of my former students who come to appreciate the importance of proper understanding of Statistics only after graduating from the University! Finally, I extend my warm gratitude to Mr. Nathan Ayesu Kwafo and Miss Amelia A. Agbekudzi of the Department of Statistics, University of Ghana, Legon, for painstakingly typing the manuscript to its present readable form.

University of Ghana, Legon. J.Y.OpokuAugust 2004

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PREFACE TO THE REVISED EDITION

While the First Edition of this book that came out in 2004 has greatly served the purpose for which it was intended, namely, to demystify the fear surrounding Inferential Social Statistics by students, a critical re-examination of the First Edition suggested to me the need to revise certain aspects of it. This revision is not in terms of content or coverage, but involves mainly:

Some few typographical errors; Some few infelicities in expression; and Some few computational errors that were all not detected at

the time of the preparation of the manuscript. An inclusion of an Index that was inadvertently missing in the

First Edition. A more appealing format of presentation of the text. Also, as is evident from the title of this revised edition, “A

companion for undergraduate students” has been dropped because information reaching me indicates that the book is quite useful to both undergraduate and graduate students in the Universities and other tertiary institutions, as well as researchers.

In doing this revision, what was foremost in my mind was that writers must continually appraise the works they produce and revise them when necessary. In this way, they will continue to improve upon their works, for others to also improve upon them when necessary.

J.Y. OpokuUniversity of Ghana LegonApril, 2006

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FOREWORD

At long last, the myth surrounding Inferential Social Statistics is broken! In this book, the author, who has a firm grounding in Mathematics and Social Science research takes the reader through some selected but important topics in Inferential Social Statistics in such a manner that the reader feels the author is speaking in his/her presence. The author does not make any presumptions about the reader’s knowledge in Mathematics – he uses a step-by-step approach in the handling of all topics treated that even so-called “difficult” concepts become so easy to understand that one starts wondering why such concepts were considered difficult in the first place.

The book is written in very clear and simple language and is replete with examples from Social Science research. The author’s rich experience in teaching and research is vividly brought to bear in the handling of the various topics treated. The book is destined to become the “Bible” for undergraduate and graduate students in the Social Sciences and Education preparing towards examinations in the much-feared course in Inferential Statistics and/or embarking on analysis of research data involving small samples. The author’s style of presentation indeed makes the book qualify as “Tutorials in Inferential Social statistics”.

The book is highly recommended for all undergraduate and graduate students in the Social Sciences and Education and those in allied disciplines, as well as graduate students with no formal background in Statistics.

I.G. Akar, PhDDepartment of StatisticsUniversity of GhanaLegon, Ghana.

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The title of this book, “Tutorials in Inferential Social Statistics”, conveys a particular intention of the author: To present a truly readable and student-friendly introduction to Inferential Social Statistics- one that students would trust, befriend and keep as a “bully”. Indeed, this is a handbook that will motivate students because the author has presented the basic concepts of Inferential Social Statistics in a context that demystifies the fear surrounding the subject and he has skillfully related these concepts to students’ personal experiences and the research problems they would encounter.

Throughout the text, the primary focus of the author is to present without oversimplification, the basic concepts of Inferential Social Statistics in a clear, accessible, handy and enjoyable manner so that students would read the text with or without a tutor because it is spiced now and again with practical examples and sample problems. Attention to theoretical details is done in a manner that the reader will find refreshingly different from other texts. The author who has taught Statistics for the Behavioural Sciences for nearly 20 years writes in an appealing style that makes the material he presents easy to grasp.

Reading this book is like actually sitting across from the author and becoming engaged in an engrossing and educative tutorial with him. This book would be referred to again and again like a handbook whenever a research is to be conducted or an examination is to be written. It provides a real therapy for treating the popular psychopathology- “statisticophobia” (fear of Statistics).

Finally, as former students of the author, we would like to recommend these tips to all students who are desirous in passing Inferential Statistics- attend lectures regularly, do the reading in this book, practice the exercises, attend tutorials, discuss topics with course mates, and lastly, review what you have learnt. From our experience with the author, there is no other substitute for obtaining a good grade in Inferential Statistics apart from following these recommended tips. Good luck!

Dedon Momo Kromah & Esther AmakyeFormer students of the author at theUniversity of Ghana, Legon.

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ORGANIZATION OF THE BOOK

This book, written to ideally cover about a semester’s course in Inferential Social Statistics, is organized along the following lines:

Section A consists of Chapters 1 and 2. Chapter 1 Theoretical Introduction takes the reader through some basic concepts and principles that are needed for a proper understanding of the topics treated in later Chapters. These concepts and principles include: “Summation Notation”, Levels of Measurement that are used in Social Science research and the statistical tests applicable to each level of measurement; and the concepts of standard deviation and variance of a set of data.

Chapter 2 continues with the theoretical introduction, explaining among others, concepts such as the probability of independent events and its application to research; the decision rule (level of significance) employed in hypothesis testing; the standard error of the mean; and the difference between one and two-tailed tests of significance.

Chapters 3, 4, 5, 6 and 7 (Section B) treat basic Parametric Statistical tests. The Z and t tests for a single population are treated in Chapter 3. This is followed by the t test for the difference between the means of 2 populations in Chapter 4 and the t test for matched (correlated) samples in Chapter 5. Chapter 6 introduces the reader to a very important topic: OneWay Analysis of Variance (OneWay ANOVA) and the “protected” t test method of multiple comparisons following a significant F test. Chapter 7 deals with the Pearson Product Moment Correlation Coefficient (r).

Section C discusses some selected topics in Nonparametric Statistics. These include the Spearman Rank-Order Correlation Coefficient (rs), the Chi Square (χ²) Test, the Mann-Whitney U Test, The Wilcoxon matched-pairs signed-ranks Test, and the Kruskal-Wallis one-way ANOVA by ranks. Each Chapter contains a comprehensive summary of the concepts treated in that chapter. From Chapter 3 through Chapter 12, the treatment of each statistical test is followed by one worked example and two exercises on the topic. This is intended to make the reader an active

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participant and should obviously enhance the understanding of each topic. A comprehensive recommended reading list is provided, followed by Appendices that consist of some selected useful computational formulae as Appendix 1, and some selected Statistical Tables as Appendix 2. Appendix 3 provides guidelines to the solution of the exercises set. It is hoped that the topics treated in the book will cover most data analysis required for undergraduate research projects. I wish you an enjoyable study of Inferential Social Statistics!

Author

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