Understanding Probability and Statistics: A Book of Problems

254 Pages·16.697 MB·English

by  Ruma Falk

U nderstanding
Probability and
Statistics

Understanding
Probability and
Statistics
A Book of Problems
Ruma Falk
with the cooperation of Raphael Falk
The Hebrew University of Jerusalem
Department of Psychology and School of Education
Jerusalem, Israel
LONDON AND

In memory of Leslie V. Glickman (1945-1991).
Friend, teacher, statistician.

Contents
Preface ix
I PROBLEM S TO SOLVE 1
1 Descriptive Statistics I 3
1.1 Measures Characterizing Distributions I .......................... 3
1.2 Correlation and Linear Regression...........................................16
2 Probability I 21
2.1 Events, Operations, and Event Probabilities.........................21
2.2 Problems Involving Combinatorics..........................................27
2.3 Conditional Probabilities and Dependence/Independence
between Events...........................................................................36
2.4 Bayes’ Theorem..........................................................................48
2.5 Probability Distributions and Expectations..........................59
3 Reasoning across Domains 69
3.1 Statistics, Probability, and Inference ....................................69
II M ULTIPLE-CHOICE PROBLEM S 85
4 Descriptive Statistics II 87
4.1 Scales of Measurement..............................................................87
4.2 Measures Characterizing Distributions I I .............................91
4.3 Transformed Scores, Association, and Linear Regression . 101
4.4 In Retrospect............................................................................110
vii
viii Contents
5 Probability II 111
6 Normal Distribution, Sampling Distributions, and
Inference 121
III ANSW ERS 131
Answers for Part I: Problems to Solve 133
Chapter 1. Descriptive Statistics I ..................................................133
1.1 Measures Characterizing Distributions I .....................133
1.2 Correlation and Linear Regression..................................142
Chapter 2. Probability I ..................................................................144
2.1 Events, Operations, and Event Probabilities...............144
2.2 Problems Involving Combinatorics..................................147
2.3 Conditional Probabilities and
Dependence/Independence between Events . . . . 158
2.4 Bayes’ Theorem..................................................................170
2.5 Probability Distributions and Expectations..................189
Chapter 3. Reasoning across Domains ........................................203
3.1 Statistics, Probability, and Inference ...........................203
Answers for Part II: Multiple-Choice Problems 217
Chapter 4. Descriptive Statistics I I ...............................................217
4.1 Scales of Measurement.....................................................217
4.2 Measures Characterizing Distributions I I .....................217
4.3 Transformed Scores, Association, and Linear
Regression.....................................................................218
4.4 In Retrospect.....................................................................218
Chapter 5. Probability II..................................................................218
Chapter 6. Normal Distribution, Sampling Distributions, and
Inference.....................................................................................219
References 221
Index 232
Preface
We certainly think we know whether we understand something or not;
and most of us have a fairly deep-rooted belief that it matters.
Skemp (1971, p. 14)
Statistics and probability are often perceived as threats by students,
particularly those in the humanities and the social sciences and not
so rarely in the natural sciences as well. The very mention of statis­
tics evokes the notorious anxiety associated with mathematics. Fur­
thermore, dealing with uncertainty adds a disconcerting dimension to
statistics. When you solve a mathematical problem, hard as it may be,
you are rewarded by attaining certainty, but in working out a statistical
problem, all your efforts may end up in an average or a probability value
which is incapable of telling what will happen next. At best, you may
learn what will happen in the long run, and that too, only in terms of
proportions. Some students find this state of affairs quite frustrating.
Without getting into the problem of the roots of math-anxiety and
statistics-phobia and the detailed methods of overcoming them, we feel
that the best way to allay these fears is by experiencing the revelation
of understanding. One safe way toward this goal is to actively tackle
problems concerning the concepts you learn. Nothing is more reassuring
than coping with a problem by going back to the definitions of the
concepts involved and manipulating them by the rules of the game.
One solved problem, worked out on your own, may be worth tens of
pages of passive reading.
When some of our more diligent students ask for extra reading, we
often discourage these good intentions and recommend instead that
they use their energy for independent exploration of problems. Even
reading a statistical text is best accomplished when conducted as a
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