Description

Please discuss, elaborate, and reflect on the following from chapters 4 & 5. Below listed are the important topics that you have to include in your discussions. Give examples and elaborate on the applications of the topic.

Attached chapter 7 and 8 topics

Attached Textbook

12th Edition

Exploring Statistics

Tales of Distributions

Chris Spatz

Outcrop Publishers

Conway, Arkansas

Exploring Statistics: Tales of Distributions

12th Edition

Chris Spatz

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234567

24 23 22 21 20

About The Author

About The Author

Chris Spatz is at Hendrix College where he twice served as chair of

the Psychology Department. Dr. SpatzÃ¢â‚¬â„¢s undergraduate education

was at Hendrix, and his PhD in experimental psychology is from

Tulane University in New Orleans. He subsequently completed

postdoctoral fellowships in animal behavior at the University of

California, Berkeley, and the University of Michigan. Before

returning to Hendrix to teach, Spatz held positions at The University

of the South and the University of Arkansas at Monticello.

Spatz served as a reviewer for the journal Teaching of Psychology

for more than 20 years. He co-authored a research methods textbook,

wrote several chapters for edited books, and was a section editor for the

Encyclopedia of Statistics in Behavioral Science.

In addition to writing and publishing, Dr. Spatz enjoys the outdoors,

especially canoeing, camping, and gardening. He swims several times

a week (mode = 3). Spatz has been an opponent of high textbook prices for years, and he is

happy to be part of a new wave of authors who provide high-quality textbooks to students at

affordable prices.

v

vi

Dedication

With love and affection,

this textbook is dedicated to

Thea Siria Spatz, Ed.D., CHES

Brief Contents

Brief Contents

Preface xiv

1 Introduction 1

2 Exploring Data: Frequency Distributions and Graphs 29

3 Exploring Data: Central Tendency 45

4 Exploring Data: Variability 59

5 Other Descriptive Statistics 77

6 Correlation and Regression 94

7 Theoretical Distributions Including the Normal Distribution 127

8 Samples, Sampling Distributions, and Confidence Intervals 150

9 Effect Size and NHST: One-Sample Designs 175

10 Effect Size, Confidence Intervals, and NHST:

Two-Sample Designs 200

11 Analysis of Variance: Independent Samples 231

12 Analysis of Variance: Repeated Measures 259

13 Analysis of Variance: Factorial Design 271

14 Chi Square Tests 303

15 More Nonparametric Tests 328

16 Choosing Tests and Writing Interpretations 356

Appendixes

A Getting Started 371

B Grouped Frequency Distributions and Central Tendency 376

C Tables 380

D Glossary of Words 401

E Glossary of Symbols 405

F Glossary of Formulas 407

G Answers to Problems 414

References 466

Index 472

vii

viii

Contents

Contents

Preface

xiv

chapter 1 Introduction 1

Disciplines That Use Quantitative Data 5

What Do You Mean, Ã¢â‚¬Å“StatisticsÃ¢â‚¬Â? 6

Statistics: A Dynamic Discipline 8

Some Terminology 9

Problems and Answers 12

Scales of Measurement 13

Statistics and Experimental Design 16

Experimental Design Variables 17

Statistics and Philosophy 20

Statistics: Then and Now 21

How to Analyze a Data Set 22

Helpful Features of This Book 22

Computers, Calculators, and Pencils 24

Concluding Thoughts 25

Key Terms 27

Transition Passage to Descriptive Statistics 28

chapter 2 Exploring Data: Frequency Distributions

and Graphs 29

Simple Frequency Distributions 31

Grouped Frequency Distributions 33

Graphs of Frequency Distributions 35

Describing Distributions 39

Contents

The Line Graph 41

More on Graphics 42

A Moment to Reflect 43

Key Terms 44

chapter 3 Exploring Data: Central Tendency

45

Measures of Central Tendency 46

Finding Central Tendency of Simple Frequency Distributions 49

When to Use the Mean, Median, and Mode 52

Determining Skewness From the Mean and Median 54

The Weighted Mean 55

Estimating Answers 56

Key Terms 58

chapter 4 Exploring Data: Variability

59

Range 61

Interquartile Range 61

Standard Deviation 63

Standard Deviation as a Descriptive Index of Variability

Ã…Â as an Estimate of ÃÆ’ 69

Variance 73

Statistical Software Programs 74

Key Terms 76

chapter 5 Other Descriptive Statistics

Describing Individual Scores 78

Boxplots 82

Effect Size Index 86

The Descriptive Statistics Report 89

Key Terms 92

77

Transition Passage to Bivariate Statistics

chapter 6 Correlation and Regression

Bivariate Distributions 96

Positive Correlation 96

Negative Correlation 99

Zero Correlation 101

Correlation Coefficient 102

Scatterplots 106

64

94

93

ix

x

Contents

Interpretations of r 106

Uses of r 110

Strong Relationships but Low Correlation Coefficients

Other Kinds of Correlation Coefficients 115

Linear Regression 116

The Regression Equation 117

Key Terms 124

What Would You Recommend? Chapters 2-6 125

112

Transition Passage to Inferential Statistics

126

chapter 7 Theoretical Distributions Including the

Normal Distribution 127

Probability 128

A Rectangular Distribution 129

A Binomial Distribution 130

Comparison of Theoretical and Empirical Distributions 131

The Normal Distribution 132

Comparison of Theoretical and Empirical Answers 146

Other Theoretical Distributions 146

Key Terms 147

Transition Passage to the Analysis of Data From

Experiments 149

chapter 8 Samples, Sampling Distributions, and

Confidence Intervals 150

Random Samples 152

Biased Samples 155

Research Samples 156

Sampling Distributions 157

Sampling Distribution of the Mean 157

Central Limit Theorem 159

Constructing a Sampling Distribution When ÃÆ’ Is Not Available

The t Distribution 165

Confidence Interval About a Population Mean 168

Categories of Inferential Statistics 172

Key Terms 173

164

Contents

Transition Passage to Null Hypothesis Significance

Testing 174

chapter 9 Effect Size and NHST: One-Sample Designs

175

Effect Size Index 176

The Logic of Null Hypothesis Significance Testing (NHST) 179

Using the t Distribution for Null Hypothesis Significance Testing 182

A Problem and the Accepted Solution 184

The One-Sample t Test 186

An Analysis of Possible Mistakes 188

The Meaning of p in p < .05 191
One-Tailed and Two-Tailed Tests 192
Other Sampling Distributions 195
Using the t Distribution to Test the Significance of a Correlation
Coefficient 195
t Distribution Background 197
Why .05? 198
Key Terms 199
chapter 10 Effect Size, Confidence
Intervals, and NHST: Two-Sample Designs 200
A Short Lesson on How to Design an Experiment 201
Two Designs: Paired Samples and Independent Samples
Degrees of Freedom 206
Paired-Samples Design 208
Independent-Samples Design 212
The NHST Approach 217
Statistical Significance and Importance 222
Reaching Correct Conclusions 222
Statistical Power 225
Key Terms 228
What Would You Recommend? Chapters 7-10 229
Transition Passage to More Complex Designs
202
230
xi
xii
Contents
chapter 11 Analysis of Variance: Independent Samples
231
Rationale of ANOVA 233
More New Terms 240
Sums of Squares 240
Mean Squares and Degrees of Freedom 245
Calculation and Interpretation of F Values Using the F Distribution
Schedules of ReinforcementÃ¢â‚¬â€A Lesson in Persistence 248
Comparisons Among Means 250
Assumptions of the Analysis of Variance 254
Random Assignment 254
Effect Size Indexes and Power 255
Key Terms 258
chapter 12 Analysis of Variance: Repeated Measures
246
259
A Data Set 260
Repeated-Measures ANOVA: The Rationale 261
An Example Problem 262
Tukey HSD Tests 265
Type I and Type II Errors 266
Some Behind-the-Scenes Information About Repeated-Measures ANOVA
267
Key Terms 270
chapter 13 Analysis of Variance: Factorial Design
Factorial Design 272
Main Effects and Interaction 276
A Simple Example of a Factorial Design 282
Analysis of a 2 Ãƒâ€” 3 Design 291
Comparing Levels Within a FactorÃ¢â‚¬â€Tukey HSD Tests
Effect Size Indexes for Factorial ANOVA 299
Restrictions and Limitations 299
Key Terms
301
297
Transition Passage to Nonparametric Statistics
chapter 14 Chi Square Tests
303
The Chi Square Distribution and the Chi Square Test
Chi Square as a Test of Independence 307
Shortcut for Any 2 Ãƒâ€” 2 Table 310
Effect Size Indexes for 2 Ãƒâ€” 2 Tables 310
Chi Square as a Test for Goodness of Fit 314
271
305
302
Contents
Chi Square With More Than One Degree of Freedom
Small Expected Frequencies 321
When You May Use Chi Square 324
Key Terms 327
chapter 15 More Nonparametric Tests
316
328
The Rationale of Nonparametric Tests 329
Comparison of Nonparametric to Parametric Tests 330
Mann-Whitney U Test 332
Wilcoxon Signed-Rank T Test 339
Wilcoxon-Wilcox Multiple-Comparisons Test 344
Correlation of Ranked Data 348
Key Terms 353
What Would You Recommend? Chapters 11-15 353
chapter 16 Choosing Tests and Writing Interpretations
A Review 356
My (Almost) Final Word 357
Future Steps 358
Choosing Tests and Writing Interpretations
Key Term 368
Appendixes
A
B
C
D
E
F
G
359
Getting Started
371
Grouped Frequency Distributions and Central
Tendency 376
Tables
380
Glossary of Words
401
Glossary of Symbols
405
Glossary of Formulas
407
Answers to Problems
414
References
466
Index
472
356
xiii
xiv
Preface
Preface
Even if our statistical appetite is far from keen, we all of us should like to know enough
to understand, or to withstand, the statistics that are constantly being thrown at us in
print or conversationÃ¢â‚¬â€much of it pretty bad statistics. The only cure for bad statistics is
apparently more and better statistics. All in all, it certainly appears that the rudiments of
sound statistical sense are coming to be an essential of a liberal education.
Ã¢â‚¬â€œ Robert Sessions Woodworth
Exploring Statistics: Tales of Distributions (12th edition) is a textbook for a one-term statistics
course in the social or behavioral sciences, education, or an allied health/nursing field.
Its focus is conceptualization, understanding, and interpretation, rather than computation.
Designed to be comprehensible and complete for students who take only one statistics course,
it also includes elements that prepare students for additional statistics courses. For example,
basic experimental design terms such as independent and dependent variables are explained
so students can be expected to write fairly complete interpretations of their analyses. In many
places, the student is invited to stop and think or do a thought exercise. Some problems ask
the student to decide which statistical technique is appropriate. In sum, this bookÃ¢â‚¬â„¢s approach is
in tune with instructors who emphasize critical thinking in their course.
This textbook has been remarkably successful for more than 40 years. Students,
professors, and reviewers have praised it. A common refrain is that the book has a
conversational, narrative style that is engaging, especially for a statistics text. Other features
that distinguish this textbook from others include the following:
Ã¢â‚¬Â¢ Data sets are approached with an attitude of exploration.
Ã¢â‚¬Â¢ Changes in statistical practice over the years are acknowledged, especially the recent
emphasis on effect sizes and confidence intervals.
Ã¢â‚¬Â¢ Criticism of null hypothesis significance testing (NHST) is explained.
Ã¢â‚¬Â¢ Examples and problems represent a variety of disciplines and everyday life.
Ã¢â‚¬Â¢ Most problems are based on actual studies rather than fabricated scenarios.
Ã¢â‚¬Â¢ Interpretation is emphasized throughout.
Ã¢â‚¬Â¢ Problems are interspersed within a chapter, not grouped at the end.
Ã¢â‚¬Â¢ Answers to all problems are included.
Ã¢â‚¬Â¢ Answers are comprehensively explainedÃ¢â‚¬â€over 50 pages of detail.
Ã¢â‚¬Â¢ A final chapter, Choosing Tests and Writing Interpretations, requires active responses to
comprehensive questions.
Preface
Ã¢â‚¬Â¢ Effect size indexes are treated as important descriptive statistics, not add-ons to NHST.
Ã¢â‚¬Â¢ Important words and phrases are defined in the margin when they first occur.
Ã¢â‚¬Â¢ Objectives, which open each chapter, serve first for orientation and later as review
items.
Ã¢â‚¬Â¢ Key Terms are identified for each chapter.
Ã¢â‚¬Â¢ Clues to the Future alert students to concepts that come up again.
Ã¢â‚¬Â¢ Error Detection boxes tell ways to detect mistakes or prevent them.
Ã¢â‚¬Â¢ Transition Passages alert students to a change in focus in chapters that follow.
Ã¢â‚¬Â¢ Comprehensive Problems encompass all (or most) of the techniques in a chapter.
Ã¢â‚¬Â¢ What Would You Recommend? problems require choices from among techniques in
several chapters.
For this 12th edition, I increased the emphasis on effect sizes and confidence intervals,
moving them to the front of Chapter 9 and Chapter 10. The controversy over NHST is
addressed more thoroughly. Power gets additional attention. Of course, examples and
problems based on contemporary data are updated, and there are a few new problems. In
addition, a helpful Study Guide to Accompany Exploring Statistics (12th edition) was written
by Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, all of Hendrix College. The study
guide is available online at exploringstatistics.com.
Students who engage in this book and their course can expect to:
Ã¢â‚¬Â¢ Solve statistical problems
Ã¢â‚¬Â¢ Understand and explain statistical reasoning
Ã¢â‚¬Â¢ Choose appropriate statistical techniques for common research designs
Ã¢â‚¬Â¢ Write explanations that are congruent with statistical analyses
After many editions with a conventional publisher, Exploring Statistics: Tales of
Distributions is now published by Outcrop Publishers. As a result, the price of the print
edition is about one-fourth that of the 10th edition. Nevertheless, the authorship and quality of
earlier editions continue as before.
xv
xvi
Preface
Acknowledgments
The person I acknowledge first is the person who most deserves acknowledgment. And for the
11th and 12th editions, she is especially deserving. This book and its accompanying publishing company,
Outcrop Publishers, would not exist except for Thea Siria Spatz, encourager, supporter, proofreader, and
cheer captain. This edition, like all its predecessors, is dedicated to her.
Kevin Spatz, manager of Outcrop Publishers, directed the distribution of the 11th edition,
advised, week by week, and suggested the cover design for the 12th edition. Justin Murdock now serves
as manager, continuing the tradition that Kevin started. Tina Haggard of Fingertek Web Design created
the bookÃ¢â‚¬â„¢s website, the textÃ¢â‚¬â„¢s ebook, and the online study guide. She provided advice and solutions for
many problems. Thanks to Jill Schmidlkofer, who edited the extensive answer section again for this
edition. Emily Jones Spatz created new drawings for the text. IÃ¢â‚¬â„¢m particularly grateful to Grace Oxley for
a cover design that conveys exploration, and to Liann Lech, who copyedited for clarity and consistency.
WalsworthÃ‚Â® turned a messy collection of files into a handsome bookÃ¢â‚¬â€thank you Nathan Stufflebean
and Dennis Paalhar. Others who were instrumental in this edition or its predecessors include Jon Arms,
Ellen Bruce, Mary Kay Dunaway, Bob Eslinger, James O. Johnston, Roger E. Kirk, Rob Nichols, Jennifer
Peszka, Mark Spatz, and Selene Spatz. I am especially grateful to Hendrix College and my Hendrix
colleagues for their support over many years, and in particular, to Lindsay Kennedy, Jennifer Peszka, and
Leslie Zorwick, who wrote the study guide that accompanies the text.
This textbook has benefited from perceptive reviews and significant suggestions by some 90
statistics teachers over the years. For this 12th edition, I particularly thank
Jessica Alexander, Centenary College
Lindsay Kennedy, Hendrix College
Se-Kang Kim, Fordham University
Roger E. Kirk, Baylor University
Kristi Lekies, The Ohio State University
Jennifer Peszka, Hendrix College
Robert Rosenthal, University of California, Riverside
IÃ¢â‚¬â„¢ve always had a touch of the teacher in meÃ¢â‚¬â€as an older sibling, a parent, a professor, and now
a grandfather. Education is a first-class task, in my opinion. I hope this book conveys my enthusiasm for
it. (By the way, if you are a student who is so thorough as to read even the acknowledgments, you should
know that I included phrases and examples in a number of places that reward your kind of diligence.)
If you find errors in this book, please report them to me at spatz@hendrix.edu. I will post
corrections at the bookÃ¢â‚¬â„¢s website: exploringstatistics.com.
127
Theoretical Distributions
Including the Normal
Distribution
CHAPTER
7
OBJECTIVES FOR CHAPTER 7
After studying the text and working the problems in this chapter, you should be
able to:
1. Distinguish between theoretical and empirical distributions
2. Distinguish between theoretical and empirical probability
3. Describe the rectangular distribution and the binomial distribution
4. Find the probability of certain events from knowledge of the theoretical distribution
of those events
5. List the characteristics of the normal distribution
6. Find the proportion of a normal distribution that lies between two scores
7. Find the scores between which a certain proportion of a normal distribution falls
8. Find the number of cases associated with a particular proportion of a normal
distribution
THIS CHAPTER HAS more figures than any other chapter, almost one per page. The
reason for all these figures is that they are the best way I know to convey ideas about
theoretical distributions and probability. So, please examine these figures carefully,
making sure you understand what each part means. When you are working problems,
drawing your own pictures is a big help.
IÃ¢â‚¬â„¢ll begin by distinguishing between empirical distributions and theoretical
distributions. In Chapter 2, you learned to arrange scores in frequency distributions. The
scores you worked with were selected because they were representative of
Empirical distribution
scores from actual research. Distributions of observed scores are empirical Scores that come from
distributions.
observations.
This chapter has a heavy emphasis on theoretical distributions. Like
Theoretical distribution
the empirical distributions in Chapter 2, a theoretical distribution is a Hypothesized scores based
presentation of all the scores, usually presented as a graph. Theoretical on mathematical formulas and
logic.
distributions, however, are based on mathematical formulas and logic rather
than on empirical observations.
Theoretical distributions are used in statistics to determine probabilities. When there
is a correspondence between an empirical distribution and a theoretical distribution, you
can use the theoretical distribution to arrive at probabilities about future empirical events.
Probabilities, as you know, are quite helpful in reaching decisions.
128
Chapter 7
This chapter covers three theoretical distributions: rectangular, binomial, and normal.
Rectangular and binomial distributions are used to illustrate probability more fully and to
establish some points that are true for all theoretical distributions. The third distribution, the
normal distribution, will occupy the bulk of your time and attention in this chapter.
Probability
You already have some familiarity with the concept of probability. You know, for example, that
probability values range from .00 (there is no possibility that an event will occur) to 1.00 (the
event is certain to happen).
In probability language, events are often referred to as Ã¢â‚¬Å“successesÃ¢â‚¬Â or Ã¢â‚¬Å“failures.Ã¢â‚¬Â To
calculate the probability of a success using the theoretical approach, first enumerate all the ways
a success can occur. Then enumerate all the events that can occur (whether successes or failures).
Finally, form a ratio with successes in the numerator and total events in the denominator. This
fraction, changed to a decimal, is the theoretical probability of a success. For example, with
coin flipping, the theoretical probability of Ã¢â‚¬Å“headÃ¢â‚¬Â is .50. A head is a success and it can occur
in only one way. The total number of possible outcomes is two (head and tail), and the ratio 1/2
is .50. In a similar way, the probability of rolling a six on a die is 1/6 = .167. For playing cards,
the probability of drawing a jack is 4/52 = .077.
The empirical approach to finding probability involves observing actual events, some of
which are successes and some of which are failures. The ratio of successes to total events
produces a probability, a decimal number between .00 and 1.00. To find an empirical probability,
you use observations rather than logic to get the numbers.1
What is the probability of particular college majors? This probability question can be
answered using numbers from Figure 2.5 and the fact that 1,921,000 baccalaureate degrees
were granted in 2015Ã¢â‚¬â€œ2016. Choose the major you are interested in and label the frequency
of that major number of successes. Divide that by 1,921,000, the number of events. The result
answers the probability question. If the major in question is sociology, then 26,000/1,921,000
= .01. For English, the probability is 43,000/1,921,000 = .02.2 Now hereÃ¢â‚¬â„¢s a question for you to
answer for yourself. Were these probabilities determined theoretically or empirically?
The rest of this chapter will emphasize theoretical distributions and theoretical probability.
You will work with coins and cards next, but before you are finished, I promise you a much
wider variety of applications.
1
2
The empirical probability approach is sometimes called the relative frequency approach.
If I missed doing the arithmetic for the major you are interested in, I hope youÃ¢â‚¬â„¢ll do it for yourself.
Theoretical Distributions Including Normal Distribution
A Rectangular Distribution
To show you the relationship between theoretical distributions and theoretical
Rectangular distribution
probabilities, IÃ¢â‚¬â„¢ll use a theoretical distribution based on a deck of ordinary Distribution in which all scores
playing cards. Figure 7.1 is a histogram that shows the distribution of types have the same frequency.
of cards. There are 13 kinds of cards, and the frequency of each card is 4.
This theoretical curve is a rectangular distribution. (The line that encloses
a histogram or frequency polygon is called a curve, even if it is straight.) The number in the area
above each card is the probability of obtaining that card in a chance draw from the deck. That
theoretical probability (.077) was obtained by dividing the number of cards that represent the
event (4) by the total number of cards (52).
F I G U R E 7 . 1 Theoretical distribution of 52 draws from a deck of playing cards
Probabilities are often stated as Ã¢â‚¬Å“chances in a hundred.Ã¢â‚¬Â The expression p = .077 means
that there are 7.7 chances in 100 of the event occurring. Thus, from Figure 7.1, you can tell at a
glance that there are 7.7 chances in 100 of drawing an ace from a deck of cards. This knowledge
might be helpful in some card games.
With this theoretical distribution, you can determine other probabilities. Suppose you want
to know your chances of drawing a face card or a 10. These are the shaded events in Figure 7.1.
Simply add the probabilities associated with a 10, jack, queen, and king. Thus, .077 + .077 +
.077 + .077 = .308. This knowledge might be helpful in a game of blackjack, in which a face
card or a 10 is an important event (and may even signal Ã¢â‚¬Å“successÃ¢â‚¬Â).
In Figure 7.1, there are 13 kinds of events, each with a probability of .077. It is not
surprising that when you add them all up [(13)(.077)], the result is 1.00. In addition to the
probabilities adding up to 1.00, the areas add up to 1.00. That is, by conventional agreement,
the area under the curve is taken to be 1.00. With this arrangement, any statement about area is
also a statement about probability. (If you like to verify things for yourself, youÃ¢â‚¬â„¢ll find that each
129
130
Chapter 7
slender rectangle has an area that is .077 of the area under the curve.) Of the total area under
the curve, the proportion that signifies ace is .077, and that is also the probability of drawing an
ace from the deck.3
clue to the future
The probability of an event or a group of events corresponds to the area of the theoretical
distribution associated with the event or group of events. This idea will be used
throughout this book.
PROBLEMS
7.1. What is the probability of drawing a card that falls between 3 and jack, excluding both?
7.2. If you drew a card at random, recorded the result, and replaced the card in the deck, how
many 7s would you expect in 52 draws?
7.3. What is the probability of drawing a card that is higher than a jack or lower than a 3?
7.4. If you made 78 draws from a deck, replacing each card, how many 5s and 6s would you
expect?
A Binomial Distribution
The binomial distribution is another example of a theoretical distribution.
Suppose you take three new quarters and toss them into the air. What is the
probability that all three will come up heads? As you may know, the answer
is found by multiplying together the probabilities of each of the independent
events. For each coin, the probability of a head is 1/2 so the probability that
all three will be heads is (1/2)(1/2)(1/2) = 1/8 = .1250.
Here are two other questions about tossing those three coins. What is the probability of two
heads? What is the probability of one head or zero heads? You could answer these questions
easily if you had a theoretical distribution of the probabilities. HereÃ¢â‚¬â„¢s how to construct one.
Start by listing, as in Table 7.1, the eight possible outcomes of tossing the three quarters into the
air. Each of these eight outcomes is equally likely, so the probability for any one of them is 1/8
= .1250. There are three outcomes in which two heads appear, so the probability of two heads is
.1250 + .1250 + .1250 = .3750. The probability .3750 is the answer to the first question. Based
on Table 7.1, I constructed Figure 7.2, which is the theoretical distribution of probabilities you
need. You can use it to answer Problems 7.5 and 7.6, which follow.
Binomial distribution
Distribution of the frequency of
events that can have only two
possible outcomes.
In gambling situations, uncertainty is commonly expressed in odds. The expression Ã¢â‚¬Å“odds of 5:1Ã¢â‚¬Â means that there
are five ways to fail and one way to succeed; 3:2 means three ways to fail and two ways to succeed. The odds of
drawing an ace are 12:1. To convert odds to a probability of success, divide the second number by the sum of the two
numbers.
3
Theoretical Distributions Including Normal Distribution
T A B L E 7 . 1 All possible outcomes when three coins are tossed
Outcomes
Heads, heads, heads
Heads, heads, tails
Heads, tails, heads
Tails, heads, heads
Heads, tails, tails
Tails, head, tails
Tails, Tails, head
Tails, tails, tails
Number of heads
Probability of outcome
3
2
2
2
1
1
1
0
.1250
.1250
.1250
.1250
.1250
.1250
.1250
.1250
F I G U R E 7 . 2 A theoretical binomial distribution showing the number of
heads when three coins are tossed
PROBLEMS
7.5. If you toss three coins into the air, what is the probability of a success if success is (a) either
one head or two heads? (b) all heads or all tails?
7.6. If you throw the three coins into the air 16 times, how many times would you expect to
find zero heads?
Comparison of Theoretical and Empirical Distributions
I have carefully called Figures 7.1 and 7.2 theoretical distributions. A theoretical distribution
may not reflect exactly what would happen if you drew cards from an actual deck of playing
cards or tossed quarters into the air. Actual results could be influenced by lost or sticky cards,
sleight of hand, uneven surfaces, or chance deviations. Now letÃ¢â‚¬â„¢s turn to the empirical question
of what a frequency distribution of actual draws from a deck of playing cards looks like. Figure
7.3 is a histogram based on 52 draws from a used deck shuffled once before each draw.
131
132
Chapter 7
F I G U R E 7 . 3 Empirical frequency distribution of 52 draws from a deck of
playing cards
As you can see, Figure 7.3 is not exactly like Figure 7.1. In this case, the differences
between the two distributions are due to chance or worn cards and not to lost cards or sleight of
hand (at least not conscious sleight of hand). Of course, if I made 52 more draws from the deck
and constructed a new histogram, the picture would probably be different from both Figures 7.3
and 7.1. However, if I continued, drawing 520 or 5200 or 52,000 times, and only chance was
at work, the curve would be practically flat on the top; that is, the empirical curve would look
like the theoretical curve.
The major point here is that a theoretical curve represents the Ã¢â‚¬Å“best estimateÃ¢â‚¬Â of how the
events would actually occur. As with all estimates, a theoretical curve may produce predictions
that vary from actual observations, but in the world of real events, it is better than any other
estimate.
In summary, then, a theoretical distribution is one based on logic and mathematics rather
than on observations. It shows you the probability of each event that is part of the distribution.
When it is similar to an empirical distribution, the probability figures obtained from the
theoretical distribution are accurate predictors of actual events.
The Normal Distribution
Normal distribution
A bell-shaped, theoretical
distribution that predicts the
frequency of occurrence of
chance events.
One theoretical distribution has proved to be extremely valuableÃ¢â‚¬â€the
normal distribution. With contributions from Abraham de Moivre (1667Ã¢â‚¬â€œ
1754) and Pierre-Simon Laplace (1749Ã¢â‚¬â€œ1827), Carl Friedrich Gauss (1777Ã¢â‚¬â€œ
1855) worked out the mathematics of the curve and used it to assign precise
probabilities to errors in astronomy observations (Wight & Gable, 2005).
Theoretical Distributions Including Normal Distribution
Because the Gaussian curve was such an accurate picture of the effects of random variation,
early writers referred to the curve as the law of error. (In statistics, error means random
variation.) At the end of the 19th century, Francis Galton called the curve the normal distribution
(David, 1995). Perhaps Galton chose the word normal based on the Latin adjective normalis,
which means built with a carpenterÃ¢â‚¬â„¢s square (and therefore exactly right). Certainly, there were
statisticians during the 19th century who mistakenly believed that if data were collected without
any mistakes, the form of the distribution would be what is today called the normal distribution.
One of the early promoters of the normal curve was Adolphe Quetelet (KA-tle) (1796Ã¢â‚¬â€œ
1874), a Belgian who showed that many social and biological measurements are distributed
normally. Quetelet, who knew about the Ã¢â‚¬Å“law of errorÃ¢â‚¬Â from his work as an astronomer,
presented tables showing the correspondence between measurements such as height and chest
size and the normal curve. His measure of starvation and obesity was weight divided by height.
This index was a precursor of todayÃ¢â‚¬â„¢s BMI (body mass index). During the 19th century, Quetelet
was widely influential (Porter, 1986). Florence Nightingale, his friend and a pioneer in using
statistical analyses to improve health care, said that Quetelet was Ã¢â‚¬Å“the founder of the most
important science in the worldÃ¢â‚¬Â (Cook, 1913, p. 238 as quoted in Maindonald & Richardson,
2004). QueteletÃ¢â‚¬â„¢s work also gave Francis Galton the idea that characteristics we label Ã¢â‚¬Å“geniusÃ¢â‚¬Â
could be measured, an idea that led to the concept of correlation.4
Although many measurements are distributed approximately normally, it is not the case
that data Ã¢â‚¬Å“shouldÃ¢â‚¬Â be distributed normally. This unwarranted conclusion has been reached by
some scientists in the past.
Finally, the theoretical normal curve has an important place in statistical theory. This
importance is quite separate from the fact that empirical frequency distributions often correspond
closely to the normal curve.
Description of the Normal Distribution
Figure 7.4 is a normal distribution. It is a bell-shaped, symmetrical, theoretical distribution
based on a mathematical formula rather than on empirical observations. (Even so, if you peek
ahead to Figures 7.7, 7.8, and 7.9, you will see that empirical curves often look similar to this
theoretical distribution.) When the theoretical curve is drawn, the y-axis is sometimes omitted.
On the x-axis, z scores are used as the unit of measurement for the standardized normal curve.
z score
A raw score expressed in
standard deviation units.
where
Ã°Ââ€˜â€¹ = a raw score
Ã‚Âµ = the mean of the distribution
ÃÆ’ = the standard deviation of the distribution
Quetelet qualifies as a famous person: A statue was erected in his honor in Brussels, he was the first foreign member
of the American Statistical Association, and the Belgian government commemorated the centennial of his death with
a postage stamp (1974). For a short intellectual biography of Quetelet, see Faber (2005).
4
133
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Chapter 7
F I G U R E 7 . 4 The normal distribution
There are several other things to note about the normal distribution. The mean, the median,
and the mode are the same scoreÃ¢â‚¬â€the score on the x-axis where the curve peaks. If a line is
drawn from the peak to the mean score on the x-axis, the area under the curve to the left of the
line is half the total areaÃ¢â‚¬â€50%Ã¢â‚¬â€leaving half the area to the right of the line.
Asymptotic
The tails of the curve are asymptotic to the x-axis; that is, they never actually
Line that continually approaches
cross the axis but continue in both directions indefinitely, with the distance
but never reaches a specified
between the curve and the x-axis becoming less and less. Although in theory,
limit.
the curve never ends, it is convenient to think of (and to draw) the curve as
Inflection point
extending from Ã¢â‚¬â€œ3ÃÆ’ to +3ÃÆ’. (The table for the normal curve in Appendix C,
Point on a curve that separates
a concave upward arc from a
however, covers the area from Ã¢â‚¬â€œ4ÃÆ’ to +4ÃÆ’.)
concave downward arc, or vice
Another point about the normal distribution is that the two inflection
versa.
points in the curve are at exactly Ã¢â‚¬â€œ1ÃÆ’ and +1ÃÆ’. The inflection points are
where the curve is the steepestÃ¢â‚¬â€that is, where the curve changes from
bending upward to bending over. (See the points above Ã¢â‚¬â€œ1ÃÆ’ and +1ÃÆ’ in Figure 7.4 and think of
walking up, over, and down a bell-shaped hill.)
To end this introductory section, hereÃ¢â‚¬â„¢s a caution about the word normal. The antonym
for normal is abnormal. Curves that are not normal distributions, however, are definitely
not abnormal. There is nothing uniquely desirable about the normal distribution. Many nonnormal distributions are also useful to statisticians. Figure 7.1 is an example. It isnÃ¢â‚¬â„¢t a normal
distribution, but it can be very useful. Figure 7.5 shows what numbers were picked when an
instructor asked introductory psychology students to pick a number between 1 and 10. Figure
7.5 is a bimodal distribution with modes at 3 and 7. It isnÃ¢â‚¬â„¢t a normal distribution, but it will
prove useful later in this book.
Theoretical Distributions Including Normal Distribution
F I G U R E 7 . 5 Frequency distribution of choices of numbers between 1 and 10
The Normal Distribution Table
The theoretical normal distribution is used to determine the probability of an event, just as
Figure 7.1 was. Figure 7.6 is a picture of the normal curve, showing the probabilities associated
with certain areas. The figure shows that the probability of an event with a z score between 0
and 1.00 is .3413. For events with z scores of 1.00 or larger, the probability is .1587. These
probability figures were obtained from Table C in Appendix C. Turn to Table C now (p. 386)
and insert a bookmark. Table C is arranged so that you can begin with a z score (column A) and
find the following:
1. The area between the mean and the z score (column B)
2. The area from the z score to infinity (Ã¢Ë†Å¾) (column C)
F I G U R E 7 . 6 The normal distribution showing the probabilities of certain z
scores
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Chapter 7
In Table C, Column A, find the z score of 1.00. The proportion of the curve between the
mean and z = 1.00 is .3413. The proportion beyond a z score of 1.00 is .1587. Because the
normal curve is symmetrical and the area under the entire curve is 1.00, the sum of .3413 and
.1587 will make sense to you. Also, because the curve is symmetrical, these same proportions
hold for areas of the curve separated by z = Ã¢â‚¬â€œ1.00. Thus, all the proportions in Figure 7.6 were
derived by finding proportions associated with z = 1.00. DonÃ¢â‚¬â„¢t just read this paragraph; do it.
Understanding the normal curve now will pay you dividends throughout the book.
Notice that the proportions in Table C are carried to four decimal places and that I used all
of them. This is customary practice in dealing with the normal curve because you often want
two decimal places when a proportion is converted to a percentage.
PROBLEMS
7.7. In Chapter 5, you read of a professor who gave As to students with z scores of +1.50 or
higher.
a. What proportion of a class would be expected to make As?
b. What assumption must you make to find the proportion in 7.7a?
7.8. What proportion of the normal distribution is found in the following areas?
a. Between the mean and z = 0.21
b. Beyond z = 0.55
c. Between the mean and z = Ã¢â‚¬â€œ2.01
7.9. Is the distribution in Figure 7.5 theoretical or empirical?
As IÃ¢â‚¬â„¢ve already mentioned, many empirical distributions are approximately normally
distributed. Figure 7.7 shows a set of 261 IQ scores, Figure 7.8 shows the diameter of 199
ponderosa pine trees, and Figure 7.9 shows the hourly wage rates of 185,822 union truck
drivers in the middle of the last century (1944). As you can see, these distributions from diverse
fields are similar to Figure 7.4, the theoretical normal distribution. Please note that all of
these empirical distributions are based on a Ã¢â‚¬Å“largeÃ¢â‚¬Â number of observations. More than 100
observations are usually required for the curve to fill out nicely.
F I G U R E 7 . 7 Frequency distribution of IQ scores of 261 fifth-grade students
(unpublished data from J. O. Johnston)
Theoretical Distributions Including Normal Distribution
F I G U R E 7 . 8 Frequency distribution of diameters of 100-year-old
ponderosa pine trees on 1 acre, N=199 (Forbes & Meyer, 1955)
F I G U R E 7 . 9 Frequency distribution of hourly rates of union truck drivers
on July 1, 1944, N = 185,822 (U.S. Bureau of Labor Statistics, December 1944)
In this section, I made two statistical points: first, that Table C can be used to determine
areas (proportions) of a normal distribution, and second, that many empirical distributions are
approximately normally distributed.
Converting Empirical Distributions to the Standard Normal Distribution
The point of this section is that any normally distributed empirical distribution can be made
to correspond to the theoretical distribution in Table C by using z scores. If the raw scores of
an empirical distribution are converted to z scores, the mean of the z scores will be 0 and the
standard deviation will be 1. Thus, the parameters of the theoretical normal distribution (which
is also called the standardized normal distribution) are: mean = 0, standard deviation = 1.
Using z scores calculated from the raw scores of an empirical distribution, you can
determine the probabilities of empirical events such as IQ scores, tree diameters, and hourly
wages. In fact, with z scores, you can find the probabilities of any empirical events that are
distributed normally.
137
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Chapter 7
Human beings vary from one another in many ways, one of which is cognitive ability.
Carefully crafted tests such as Wechsler intelligence scales, the Stanford-Binet, and the
Wonderlic Personnel Test produce scores (commonly called IQ scores) that are reliable
measures of general cognitive ability. These tests have a mean of 100 and a standard deviation
of 15.5 The scores on IQ tests are normally distributed (Micceri, 1989).
Ryan (2008) provides some history and a summary of theories of intelligence, pointing
out that ancient Greeks and Chinese used measures of cognitive ability for important personnel
decisions. As you have already experienced, college admissions and other academic decisions
today are based on tests that measure cognitive ability.
PROBLEM
7.10. Calculate the z scores for IQ scores of
a. 55 b. 110 c. 103 d. 100
Proportion of a Population With Scores of a Particular Size or Greater
Suppose you are faced with finding out what proportion of the population has an IQ of 120 or
higher. Begin by sketching a normal curve (either in the margin or on separate paper). Note on
the baseline the positions of IQs of 100 and 120. What is your eyeball estimate of the proportion
with IQs of 120 or higher?
Look at Figure 7.10. It is a more formal version of your sketch, giving additional IQ scores
on the X axis. The proportion of the population with IQs of 120 or higher is shaded. The z score
that corresponds with an IQ of 120 is
F I G U R E 7 . 1 0 Theoretical distribution of IQ scores
Older versions of Stanford Binet tests had a standard deviation of 16. Also, as first noted by Flynn (1987), the actual
population mean IQ in many countries is well above 100.
5
Theoretical Distributions Including Normal Distribution
Table C shows that the proportion beyond z = 1.33 is .0918. Thus, you expect a proportion
of .0918, or 9.18%, of the population to have an IQ of 120 or higher. Because the size of an
area under the curve is also a probability statement about the events in that area, there are 9.18
chances in 100 that any randomly selected person will have an IQ of 120 or above. Figure 7.11
shows the proportions just determined.
F I G U R E 7 . 1 1 Proportion of the population with an IQ of 120 or higher
Table C gives the proportions of the normal curve for positive z scores only. However,
because the distribution is symmetrical, knowing that .0918 of the population has an IQ of 120
or higher tells you that .0918 has an IQ of 80 or lower. An IQ of 80 has a z score of Ã¢â‚¬â€œ1.33.
Questions of Ã¢â‚¬Å“How Many?Ã¢â‚¬Â
You can answer questions of Ã¢â‚¬Å“How many?Ã¢â‚¬Â as well as questions of proportions using the normal
distribution. Suppose 500 first-graders are entering school. How many would be expected to
have IQs of 120 or higher? You just found that 9.18% of the population would have IQs of 120
or higher. If the population is 500, calculating 9.18% of 500 gives you the number of children.
Thus, (.0918)(500) = 45.9. So 46 of the 500 first-graders would be expected to have an IQ of
120 or higher.
There are 19 more normal curve problems for you to do in the rest of this chapter. Do
you want to maximize your chances of working every one of them correctly the first time?
HereÃ¢â‚¬â„¢s how. For each problem, start by sketching a normal curve. Read the problem and write
the givens and the unknowns on your curve. Estimate the answer. Apply the z-score formula.
Compare your answer with your estimate; if they donÃ¢â‚¬â„¢t agree, decide which is in error and make
any changes that are appropriate. Confirm your answer by checking the answer in the back of
the book. (I hope you decide to go for 19 out of 19!)
error detection
Sketching a normal curve is the best way to understand a problem and avoid errors. Draw
vertical lines above the scores you are interested in. Write in proportions.
139
140
Chapter 7
PROBLEMS
7.11. For many school systems, an IQ of 70 indicates that the child may be eligible for special
education. What proportion of the general population has an IQ of 70 or less?
7.12. In a school district of 4,000 students, how many would be expected to have IQs of 70 or
less?
7.13. What proportion of the population would be expected to have IQs of 110 or higher?
7.14. Answer the following questions for 250 first-grade students.
a. How many would you expect to have IQs of 110 or higher?
b. How many would you expect to have IQs lower than 110?
c. How many would you expect to have IQs lower than 100?
Separating a Population Into Two Proportions
Instead of starting with an IQ score and calculating proportions, you could start with a
proportion and find an IQ score. For example, what IQ score is required to be in the top 10%
of the population?
My picture of this problem is shown as Figure 7.12. I began by sketching a more or less
bell-shaped curve and writing in the mean (100). Next, I separated the Ã¢â‚¬Å“top 10%Ã¢â‚¬Â portion with
a vertical line. Because I need to find a score, I put a question mark on the score axis.
With a picture in place, you can finish the problem. The next step is to look in Table C
under the column Ã¢â‚¬Å“area beyond zÃ¢â‚¬Â for .1000. It is not there. You have a choice between .0985
and .1003. Because .1003 is closer to the desired .1000, use it.6 The z score that corresponds to
a proportion of .1003 is 1.28. Now you have all the information you need to solve for X.
F I G U R E 7 . 1 2 Sketch of a theoretical distribution of IQ scores divided into
an upper 10% and a lower 90%
6
You might use interpolation (a method to determine an intermediate score) to find a more accurate z score for a
proportion of .1000. This extra precision (and labor) is unnecessary because the final result is rounded to the nearest
whole number. For IQ scores, the extra precision does not make any difference in the final answer.
Theoretical Distributions Including Normal Distribution
To begin, solve the basic z-score formula for X:
z=
XÃ¢â‚¬â€œÃ‚Âµ
ÃÆ’
Multiplying both sides by ÃÆ’ produces
(z)(ÃÆ’) = Ã°Ââ€˜â€¹ Ã¢â‚¬â€œ Ã‚Âµ
Adding Ã‚Âµ to both sides isolates X. Thus, when you need to find a score (X) associated with
a particular proportion of the normal curve, the formula is
Ã°Ââ€˜â€¹ = Ã‚Âµ + (z)(ÃÆ’)
Returning to the 10% problem and substituting numbers for the mean, the z score, and the
standard deviation, you get
X = 100 + (1.28)(15)
= 100 + 19.20
= 119.2
= 119 (IQs are usually expressed as whole numbers.)
Therefore, the minimum IQ score required to be in the top 10 % of the population is 119.
Here is a similar problem. Suppose a mathematics department wants to restrict the remedial
math course to those who really need it. The department has the scores on the math achievement
exam taken by entering freshmen for the past 10 years. The scores on this exam are distributed
in an approximately normal fashion, with Ã‚Âµ = 58 and ÃÆ’ = 12. The department wants to make the
remedial course available to those students whose mathematical achievement places them in
the bottom third of the freshman class. The question is, What score will divide the lower third
from the upper two thirds? Sketch your picture of the problem and check it against Figure 7.13.
F I G U R E 7 . 1 3 Distribution of scores on a math achievement exam
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Chapter 7
With a picture in place, the next step is to look in column C of Table C to find .3333. Again,
such a proportion is not listed. The nearest proportion is .3336, which has a z value of Ã¢â‚¬â€œ0.43.
(This time you are dealing with a z score below the mean, where all z scores are negative.)
Applying z = Ã¢â‚¬â€œ0.43, you get
Ã°Ââ€˜â€¹ = Ã‚Âµ + (z)(ÃÆ’) = 58 + (Ã¢â‚¬â€œ0.43)(12) = 52.84 = 53 points
Using the theoretical normal curve to establish a cutoff score is efficient. All you need
are the mean, the standard deviation, and confidence in your assumption that the scores are
distributed normally. The empirical alternative for the mathematics department is to sort
physically through all scores for the past 10 years, arrange them in a frequency distribution, and
identify the score that separates the bottom one third.
PROBLEMS
7.15. Mensa is an organization of people who have high IQs. To be eligible for membership,
a person must have an IQ Ã¢â‚¬Å“higher than 98% of the population.Ã¢â‚¬Â What IQ is required to
qualify?
7.16. The mean height of American women aged 20Ã¢â‚¬â€œ29 is 65.1 inches, with a standard deviation
of 2.8 inches (Statistical Abstract of the United States: 2012, 2013).
a. What height divides the tallest 5% of the population from the rest?
b. The minimum height required for women to join the U.S. Armed Forces is 58
inches. What proportion of the population is excluded?
*7.17. The mean height of American men aged 20Ã¢â‚¬â€œ29 is 70.0 inches, with a standard deviation
of 3.1 inches (Statistical Abstract of the United States: 2012, 2013).
a. The minimum height required for men to join the U.S. Armed Forces is 60 inches.
What proportion of the population is excluded?
b. What proportion of the population is taller than Napoleon Bonaparte, who was
5Ã¢â‚¬â„¢2Ã¢â‚¬Â?
7.18. The weight of many manufactured items is approximately normally distributed. For new
U.S. pennies, the mean is 2.50 grams and the standard deviation is 0.05 grams.
a. What proportion of all new pennies would you expect to weigh more than 2.59
grams?
b. What weights separate the middle 80% of the pennies from the lightest 10% and
the heaviest 10%?
Theoretical Distributions Including Normal Distribution
Proportion of a Population Between Two Scores
Table C in Appendix C can also be used to determine the proportion of the population between
two scores. For example, IQ scores in the range 90 to 110 are often labeled average. Is average
an appropriate adjective for this proportion? Well, what proportion of the population is between
90 and 110? Figure 7.14 is a picture of the problem.
F I G U R E 7 . 1 4 The normal distribution showing the IQ scores that define
the Ã¢â‚¬Å“averageÃ¢â‚¬Â range
In this problem, you must add an area on the left of the mean to an area on the right of the
mean. First, you need z scores that correspond to the IQ scores of 90 and 110:
z=
z=
90-100
15
=
110-100
=
15
-10
= -0.67
15
10
= 0.67
15
The proportion of the distribution between the mean and z = 0.67 is .2486, and, of course,
the same proportion is between the mean and z = Ã¢â‚¬â€œ0.67. Therefore, (2)(.2486) = .4972 or
49.72%. So approximately 50% of the population is classified as Ã¢â‚¬Å“average,Ã¢â‚¬Â using the Ã¢â‚¬Å“IQ = 90
to 110Ã¢â‚¬Â definition. Ã¢â‚¬Å“AverageÃ¢â‚¬Â seems appropriate.
What proportion of the population would be expected to have IQs between 70 and 90?
Figure 7.15 illustrates this question. There are two approaches to this problem. One is to find
the area from 100 to 70 and then subtract the area from 90 to 100. The other way is to find the
area beyond 90 and subtract from it the area beyond 70. IÃ¢â‚¬â„¢ll illustrate with the second approach.
The corresponding z scores are
and
z=
90-100
15
=
-10
= -0.67
15
z=
70-100
15
=
-30
= -2.00
15
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Chapter 7
The area beyond z = Ã¢â‚¬â€œ0.67 is .2514 and the area beyond z = Ã¢â‚¬â€œ2.00 is .0228. Subtracting the
second proportion from the first, you find that .2286 of the population has an IQ in the range
of 70 to 90.
F I G U R E 7 . 1 5 The normal distribution illustrating the area bounded by IQ
scores of 90 and 70
PROBLEMS
*7.19. The distribution of 800 test scores in an introduction to psychology course was
approximately normal, with Ã°ÂÅ“â€¡ = 35 and Ã°ÂÅ“Å½ = 6.
a. What proportion of the students had scores between 30 and 40?
b. What is the probability that a randomly selected student would score between 30
and 40?
7.20. Now that you know the proportion of students with scores between 30 and 40, would you
expect to find the same proportion between scores of 20 and 30? If so, why? If not, why
not?
7.21. Calculate the proportion of scores between 20 and 30. Be careful with this one; drawing
a picture is especially advised.
7.22. How many of the 800 students would be expected to have scores between 20 and 30?
Extreme Scores in a Population
The extreme scores in a distribution are important in many statistical applications. Most often,
extreme scores in either direction are of interest. For example, many applications focus on the
extreme 5% of the distribution. Thus, the upper 2Ã‚Â½% and the lower 2Ã‚Â½% receive attention.
Turn to Table C and find the z score that separates the extreme 2Ã‚Â½% of the curve from the rest.
(Of course, the z score associated with the lowest 2Ã‚Â½% of the curve will have a negative value.)
Please memorize the z score you just looked up. This number will turn up many times in future
chapters.
Here is an illustration. What two heart rates (beats per minute) separate the middle 95% of
the population from the extreme 5%? Figure 7.16 is my sketch of the problem. According to
Theoretical Distributions Including Normal Distribution
studies summarized by Milnor (1990), the mean heart rate for humans is 71 beats per minute
(bpm) and the standard deviation is 9 bpm. To find the two scores, use the formula
Ã°Ââ€˜â€¹ = Ã‚Âµ + (z)(ÃÆ’). Using the values given, plus the z score you memorized, you get the following:
Upper Score
Ã°Ââ€˜â€¹ = Ã‚Âµ + (z)(ÃÆ’)
= 71 + (1.96)(9)
= 71 + 17.6
= 88.6 or 89 bpm
Lower Score
Ã°Ââ€˜â€¹ = Ã‚Âµ Ã¢â‚¬â€œ (z)(ÃÆ’)
= 71 Ã¢â‚¬â€œ (1.96)(9)
= 71 Ã¢â‚¬â€œ 17.6
= 53.4 or 53 bpm
What do you make of these statistics? (Look and respond; my informal response follows.)
Ã¢â‚¬Å“Wow, look at how variable normal heart rates are! They range from 53 to 89 beats per
minute for 95% of us. Only 5% of us are outside this range. IÃ¢â‚¬â„¢m surprised.Ã¢â‚¬Â
F I G U R E 7 . 1 6 Sketch showing the separation of the extreme 5% of the
population from the rest
clue to the future
The idea of finding scores and proportions that are extreme in either direction will come up
again in later chapters. In particular, the extreme 5% and the extreme 1% are important.
PROBLEMS
7.23. What two IQ scores separate the extreme 1% of the population from the middle 99%? Set
this problem up using the Ã¢â‚¬Å“extreme 5%Ã¢â‚¬Â example as a model.
7.24. What is the probability that a randomly selected person has an IQ higher than 139 or lower
than 61?
7.25. Look at Figure 7.9 (p. 137) and suppose that the union leadership decided to ask for $0.85
per hour as a minimum wage. For those 185,822 workers, the mean was $0.99 with a
standard deviation of $0.17. If $0.85 per hour was established as a minimum, how many
workers would get raises?
7.26. Look at Figure 7.8 (p. 137) and suppose that a timber company decided to harvest all trees
8 inches DBH (diameter breast height) or larger from a 100-acre tract. On a 1-acre tract,
there were 199 trees with Ã°ÂÅ“â€¡ = 13.68 inches and Ã°ÂÅ“Å½ = 4.83 inches. How many harvestable
trees would be expected from 100 acres?
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Chapter 7
Comparison of Theoretical and Empirical Answers
You have been using the theoretical normal distribution to find probabilities and to calculate
scores and proportions of IQs, wages, heart rates, and other measures. Earlier in this chapter, I
claimed that if the empirical observations are distributed as a normal curve, accurate predictions
can be made. A reasonable question is, How accurate are all these predictions IÃ¢â‚¬â„¢ve just made?
A reasonable answer can be fashioned from a comparison of the predicted proportions (from
the theoretical curve) and the actual proportions (computed from empirical data). Figure 7.7 is
based on 261 IQ scores of fifth-grade public school students. You worked through examples
that produced proportions of people with IQs higher than 120, lower than 90, and between
90 and 110. These actual proportions can be compared with those predicted from the normal
distribution. Table 7.2 shows these comparisons.
TABLE 7.2 Comparison of predicted and actual proportions
IQs
Higher than 120
Lower than 90
Between 90 and 100
Predicted from
normal curve
Calculated from
actual data
Difference
.0918
.2514
.4972
.0920
.2069
.5249
.0002
.0445
.0277
As you can see by examining the Difference column of Table 7.2, the accuracy of the
predictions ranges from excellent to not so good. Some of this variation can be explained by
the fact that the mean IQ of the fifth-grade students was 101 and the standard deviation 13.4.
Both the higher mean (101, compared with 100 for the normal curve) and the lower standard
deviation (13.4, compared with 15) are due to the systematic exclusion of children with very
low IQ scores from regular public schools. Thus, the actual proportion of students with IQs
lower than 90 is less than predicted, which is because our school sample is not representative
of all 10- to 11- year-old children.
Although IQ scores are distributed approximately normally, many other scores are not.
Karl Pearson recognized this, as have others. Theodore Micceri (1989) made this point again in
an article titled, Ã¢â‚¬Å“The Unicorn, the Normal Curve, and Other Improbable Creatures.Ã¢â‚¬Â Caution is
always in order when you are using theoretical distributions to make predictions about empirical
events. However, donÃ¢â‚¬â„¢t let undue caution prevent you from getting the additional understanding
that statistics offers.
Other Theoretical Distributions
In this chapter, you learned a little about rectangular distributions and binomial distributions and
quite a bit about normal distributions. Later in this book, you will encounter other distributions
such as the t distribution, the F distribution, and the chi square distribution. (After all, the subtitle
of this book is Tales of Distributions.) In addition to the distributions in this book, mathematical
Theoretical Distributions Including Normal Distribution
statisticians have identified others, all of which are useful in particular circumstances. Some
have interesting names such as the Poisson distribution; others have complicated names such
as the hypergeometric distribution. In every case, however, a distribution is used because it
provides reasonably accurate probabilities about particular events.
PROBLEMS
7.27. For human infants born weighing 5.5 pounds or more, the mean gestation period is 268
days, which is just less than 9 months. The standard deviation is 14 days (McKeown &
Gibson, 1951). For gestation periods, what proportion is expected to last 10 months or
longer (300 days)?
7.28. The height of residential door openings in the United States is 6Ã¢â‚¬â„¢8Ã¢â‚¬Â. Use the information
in Problem 7.17 to determine the number of men among 10,000 who have to duck to enter
a room.
7.29. An imaginative anthropologist measured the stature of 100 hobbits (using the proper
English measure of inches) and found these values:
Ã¢Ë†â€˜X = 3600
Ã¢Ë†â€˜X 2 = 130,000
Assume that the heights of hobbits are normally distributed. Find Ã‚Âµ and ÃÆ’ and answer the
following questions.
a. The Bilbo Baggins Award for Adventure is 32 inches tall. What proportion of the
hobbit population is taller than the award?
b. Three hundred hobbits entered a cave that had an entrance 39 inches high. The
orcs chased them out. How many hobbits could exit without ducking?
c. Gandalf is 46 inches tall. What is the probability that he is a hobbit?
7.30. Please review the objectives at the beginning of the chapter. Can you do what is asked?
KEY TERMS
Asymptotic (p. 134)
Binomial distribution (p. 130)
Empirical distribution (p. 127)
Extreme scores (p. 144)
Inflection point (p. 134)
Normal distribution (p. 132)
Other theoretical distributions (p. 146)
Probability (p. 129)
Rectangular distribution (p. 129)
Theoretical distribution (p. 127)
z score (p. 133)
147
148
149
Transition Passage
To the Analysis of Data From Experiments
With your knowledge of descriptive statistics, the normal curve, and probability, you are
prepared to learn statistical techniques researchers use to analyze experiments. Experiments
compare treatments and are a hallmark of modern science.
Chapter 8 is about samples and sampling distributions. Sampling distributions provide
probabilities that are the foundation of all inferential statistics. The confidence interval, an
increasingly important inferential statistics technique, is described in Chapter 8.
Chapter 9 returns to the effect size index, d. Chapter 9 also covers the basics of null
hypothesis significance testing (NHST), the dominant statistical technique of the 20th and early
21st century. An NHST analysis results in a yes/no decision about a population parameter. In
Chapter 9, the parameters are population means and population correlation coefficients.
Chapter 10 describes simple experiments in which two treatments are administered.
The difference between the two population means is assessed. The size of the difference (d)
and a confidence interval about the amount of difference are calculated. The difference is also
subjected to an NHST analysis.
150
Samples, Sampling
Distributions, and
Confidence Intervals
CHAPTER
8
OBJECTIVES FOR CHAPTER 8
After studying the text and working the problems in this chapter, you should be able to:
1. Define random sample and obtain one if you are given a population of data
2. Define and identify biased sampling methods
3. Distinguish between random samples and the more common research samples
4. Define sampling distribution and sampling distribution of the mean
5. Discuss the Central Limit Theorem
6. Calculate a standard error of the mean using ÃÆ’ and N or sample data
7. Describe the effect of N on the standard error of the mean
8. Use the z score formula to find the probability that a sample mean or a sample
mean more extreme was drawn from a population with a specified mean
9. Describe the t distribution
10. Explain when to use the t distribution rather than the normal distribution
11. Calculate, interpret, and graphically display confidence intervals about population
means
PLEASE BE ESPECIALLY attentive to SAMPLING DISTRIBUTIONS, a concept that is
at the heart of inferential statistics. The subtitle of your book, Ã¢â‚¬Å“Tales of Distributions,Ã¢â‚¬Â is a
reference to sampling distributions; every chapter after this one is about some kind of sampling
distribution. Thus, you might consider this paragraph to be a SUPERCLUE to the future.
Here is a progression of ideas that lead to what a sampling distribution is and how it is
used. In Chapter 2, you studied frequency distributions of scores. If those scores are distributed
normally, you can use z scores to determine the probability of the occurrence of any particular
score (Chapter 7). Now imagine a frequency distribution, not of scores but of statistics, each
calculated from separate samples that are all drawn from the same population. This distribution
has a form, and if it is normal, you could use z scores to determine the probability of the
occurrence of any particular value of the statistic. A distribution of sample statistics is called a
sampling distribution.
As IÃ¢â‚¬â„¢ve mentioned several times, statistical techniques can be categorized as descriptive
and inferential. This is the first chapter that is entirely about inferential statistics, which, as you
probably recall, are methods that take chance factors into account when samples are used to
reach conclusions about populations. To take chance factors into account, you must understand
Samples, Sampling, Distributions, and Confidence Intervals
sampling distributions and what they tell you. As for samples, IÃ¢â‚¬â„¢ll describe methods of obtaining
them and some of the pitfalls. Thus, this chapter covers topics that are central to inferential
statistics.
Here is a story with an inferential statistics problem embedded in it. The problem can be
solved using techniques presented in this chapter.
Late one afternoon, two students were discussing the average family income of
students on their campus.
Ã¢â‚¬Å“Well, the average family income for college students is $95,800, nationwide,Ã¢â‚¬Â
said the junior, looking up from a book (Eagen et al., 2018). Ã¢â‚¬Å“IÃ¢â‚¬â„¢m sure the mean for
this campus is at least that much.Ã¢â‚¬Â
Ã¢â‚¬Å“I donÃ¢â‚¬â„¢t think so,Ã¢â‚¬Â the sophomore replied. Ã¢â‚¬Å“I know lots of students who have only
their own resources or come from pretty poor families. IÃ¢â‚¬â„¢ll bet you a dollar the mean
for students here at State U. is below the national average.Ã¢â‚¬Â
Ã¢â‚¬Å“YouÃ¢â‚¬â„¢re on,Ã¢â‚¬Â grinned the junior.
Together the two went out with a pencil and pad and asked 10 students how
much their family income was. The mean of these 10 answers was $86,000.
Now the sophomore grinned. Ã¢â‚¬Å“I told you so; the mean here is almost $10,000
less than the national average.Ã¢â‚¬Â
Disappointed, the junior immediately began to review their procedures. Rather
quickly, a light went on. Ã¢â‚¬Å“Actually, this average of $86,000 is meaninglessÃ¢â‚¬â€hereÃ¢â‚¬â„¢s
why. Those 10 students arenÃ¢â‚¬â„¢t representative of the whole student body. They are
late-afternoon students, and several of them support themselves with temporary jobs
while they go to school.
Most students are supported from home by parents who have permanent and
better-paying jobs. Our sample was no good. We need results from the whole
campus or at least from a representative sample.Ã¢â‚¬Â
To get the results from the whole student body, the two went the next day to the
director of the financial aid office, who told them that family incomes for the student
body are not public information. Sampling, therefore, was necessary.
After discussing their problem, the two students sought the advice of a friendly
statistics professor. The professor explained how to obtain a random sample of
students and suggested that 40 replies would be a practical sample size. Forty
students, selected randomly, were identified. After 3 days of phone calls, visits, and
callbacks, the 40 responses produced a mean of $91,900, or $3,900 less than the
national average.
Ã¢â‚¬Å“Pay,Ã¢â‚¬Â demanded the sophomore. Ã¢â‚¬Å“OK, OK, . . . here!Ã¢â‚¬Â
Later, the junior began to think. What about another random sample and its
mean? It would be different, and it could be higher. Maybe the mean of $91,900
was just bad luck, just a chance event. Shortly afterward, the junior confronted
the sophomore with thoughts about repeated sampling. Ã¢â‚¬Å“How do we know that the
151
152
Chapter 8
random sample we got told us the truth about the population? Like, maybe the mean
for the entire student body is $95,800. Maybe a sample with a mean of $91,900
would occur just by chance fairly often. I wonder what the chances are of getting a
sample mean of $91,900 from a population with a mean of $95,800?Ã¢â‚¬Â
Ã¢â‚¬Å“Well, that statistics professor told us that if we had any more questions to come
back and get an explanation of sampling distributions. In the meantime, why donÃ¢â‚¬â„¢t I
just apply my winnings toward a couple of ice cream cones.Ã¢â‚¬Â
The two statistical points of the story are that random samples are good (the junior paid
off only after the results were based on a random sample) and that uncertainty about random
samples can be reduced if you know about sampling distributions.
This chapter is about getting a sample, drawing a conclusion about the population the
sample came from, and knowing how much faith to put in your conclusion. Of course, there is
some peril in this. Even the best methods of sampling produce variable results. How can you be
sure that the sample you use will lead to a correct decision about its population? Unfortunately,
you cannot be absolutely sure.
To use a sample is to agree to accept some uncertainty about the results.
Fortunately, a sampling distribution allows you to measure uncertainty. If a great deal of
uncertainty exists, the sensible thing to do is to say you are uncertain. However, if there is very
little uncertainty, the sensible thing to do is to reach a conclusion about the population, even
though there is a small risk of being wrong. Reread this paragraph; it is important.
In this chapter, the first three sections are about samples. Following that, sampling
distributions are explained, and one method of drawing a conclusion about a population mean is
presented. In the final sections, I will explain the t distribution, which is a sampling distribution.
The t distribution is used to calculate a confidence interval, a statistic that provides information
about a population mean.
Random Samples
When it comes to finding out about a population, the best sample is a random
sample. In statistics, random refers to the method used to obtain the sample.
Subset of a population chosen
so that all samples of the
Any method that allows every possible sample of size N an equal chance to
specified size have an equal
be selected produces a random sample. Random does not mean haphazard or
probability of being selected.
unplanned. To obtain a random sample, you must do the following:
1. Define the population. That is, explain what numbers (scores) are in the population.
2. Identify every member of the population.1
3. Select numbers (scores) in such a way that every sample has an equal probability of
being selected.
Random Sample
1
Technically, the population consists of the measurements of the members and not the members themselves.
Samples, Sampling, Distributions, and Confidence Intervals
To illustrate, I will use the population of scores in Table 8.1, for which Ã‚Âµ = 9 and ÃÆ’ = 2.
Using these 20 scores allows us to satisfy Requirements 1 and 2. As for Requirement 3, IÃ¢â‚¬â„¢ll
describe three methods of selecting numbers so that all the possible samples have an equal
probability of being selected. For this illustration, N = 8.
T A B L E 8 . 1 20 scores used as a population; Ã‚Âµ = 9, ÃÆ’ = 2
9
10
7
12
11
9
13
10
10
5
8
8
10
9
6
6
8
10
8
11
One method of getting a random sample is to write each number in the population on a slip
of paper, put the 20 slips in a box, jumble them around, and draw out eight slips. The numbers
on the slips are a random sample. This method works fine if the slips are all the same size, they
are jumbled thoroughly, and the population has only a few members. If the population is large,
this method is tedious.
A second method requires a computer and a random sampling program. IBM SPSS has a
routine under the Data menu called Select Cases that allows you to choose either a specific N
or an approximate percentage of the population for your random sample.
A third method of selecting a random sample is to consult a table of random numbers, such
as Table B in Appendix C. IÃ¢â‚¬â„¢ll use this random numbers table method in this chapter because
it reveals some of the inner workings of random sampling. To use the table, you must first
assign an identifying number to each of the scores in the population. My version is Table 8.2.
The population scores do not have to be arranged in any order. Each score in the population is
identified by a two-digit number from 01 to 20.
T A B L E 8 . 2 Assignment of identifying numbers to a
population of scores
ID
number
Score
ID
number
Score
01
02
03
04
05
06
07
08
09
10
9
7
11
13
10
8
10
6
8
8
11
12
13
14
15
16
17
18
19
20
10
12
9
10
5
8
9
6
10
11
153
154
Chapter 8
Now turn to Appendix C, Table B (p. 382). Pick an intersection of a row and a
column. Any haphazard method will work; close your eyes and put your finger on a spot.
Suppose you found yourself at row 80, columns 15Ã¢â‚¬â€œ19 (p. 384). Find that place. Reading
horizontally, the digits are 82279. You need only two digits to identify any member of your
population, so you might as well use the first two (columns 15 and 16), which give you 8 and
2 (82). Unfortunately, 82 is larger than any of the identifying numbers, so it doesnÃ¢â‚¬â„¢t match a
score in the population, but at least you are started. From this point, you can read two-digit
numbers in any directionÃ¢â‚¬â€up, down, or sidewaysÃ¢â‚¬â€but the decision should be made before
you look at the numbers. If you decide to read down, you find 04. The identifying number
04 corresponds to a score of 13 in Table 8.2, so 13 becomes the first number in the sample.
The next identifying number is 34, which again does not correspond to a population score.
Indeed, the next 10 numbers are too large. The next usable ID number is 16, which places
an 8 in the sample. Continuing the search, you reach the bottom of the table. At this point,
you can go in any direction; I moved to the right and started back up the two outside columns
(18 and 19). The first number, 83, was too large, but the next ID number, 06, corresponded to
an 8, which went into the sample. Next, ID numbers of 15 and 20 correspond to population
scores of 5 and 11. The next number that is between 01 and 20 is 15, but when an ID number
has already been used, it should be ignored. The identifying numbers 18, 02, and 14 match
scores in the population of 6, 7, and 10. Thus, the random sample of eight consists of these
scores: 13, 8, 8, 5, 11, 6, 7, and 10.
PROBLEM
*8.1. A random sample is supposed to yield a statistic similar to the population parameter. Find
the mean of the random sample of eight numbers selected by the text.
What is this table of random numbers? In Table B (and in any set of random numbers),
the probability of occurrence of any digit from 0 to 9 at any place in the table is the same,
which is .10. Thus, you are just as likely to find 000 as 123 or 397. Incidentally, you cannot
generate random numbers out of your head. Certain sequences begin to recur, and (unless
warned) you will not include enough repetitions like 000 and 555. If warned, you produce
too many. Here are some suggestions for using a table of random numbers efficiently.
1. In the list of population scores and their ID numbers, check off the ID number when it is
chosen for the sample. This helps to prevent duplications.
2. If the population is large (more than 50), it is more efficient to get all the identifying
numbers from the table first. As you select them, put them in some rough order to help
prevent duplications. After you have all the identifying numbers, go to the population to
select the sample.
3. If the population has exactly 100 members, let 00 be the identifying number for 100.
By doing this, you can use two-digit identifying numbers, each one of which matches a
population score. This same technique works for populations of 10 or 1,000 members.
Random sampling has two uses: (a) It is the best method of sampling if you want to
Samples, Sampling, Distributions, and Confidence Intervals
generalize to a population. Generalizing to a population is addressed in the two sections that
follow: biased samples and research samples. (b) As an entirely separate use, random sampling
is the mathematical basis for creating sampling distributions, which are central to inferential
statistics.
PROBLEMS
8.2. Draw a random sample of 10 from the population in Table 8.1. Calculate Ã°Ââ€˜â€¹.
8.3. Draw a random sample with N = 12 from the following scores:
76 47 81 70 67 80 64 57 76 81
68 76 79 50 89 42 67 77 80 71
91 72 64 59 76 83 72 63 69
78 90 46 61 74 74 74 69 83
Biased Samples
A biased sample is one obtained by a method that systematically underselects
Biased sample
or overselects from certain groups in the population. Thus, with a biased
Sample selected in such a
sampling technique, every sample of a given size does not have an equal
way that not all samples from
the population have an equal
opportunity of being selected. With biased sampling techniques, you are
chance of being chosen.
much more likely to get an unrepresentative sample than you are with
random sampling.
At times, conclusions based on telephone surveys are suspect because
the sampling methods are biased. Suppose an investigator defines the population, identifies each
member, and calls a randomly selected sample. Suppose that 60% respond. Can valid results for the
population be based on those responses? Probably not. There is often good reason to suspect that the
60% who responded are different from the 40% who did not. Thus, although the population is made
up of both kinds of people, the sample reflects only one kind. Therefore, the sample is biased. The
probability of bias is particularly high if the survey elicits feelings of pride or despair or disgust or
apathy in some of the recipients.
A famous case of a biased sample occurred in a poll that was to predict the results of the 1936
election for president of the United States. The Literary Digest (a popular magazine) mailed 10
million ballots to those on its master mailing list, a list of more than 10 million people compiled from
Ã¢â‚¬Å“all telephone books in the U.S., rosters of clubs, lists of registered voters,Ã¢â‚¬Â and other sources. More
than 2 million ballots were returned, and the prediction was clear: Alf Landon by a landslide over
Franklin Roosevelt. As you may have learned, the actual results were just the opposite; Roosevelt
got 61% of the vote.
From the 10 million who had a chance to express a preference, 2 million very interested persons
had selected themselves. This 2 million had more than its proportional share of those who were
disgruntled with RooseveltÃ¢â‚¬â„¢s Depression-era programs. The 2 million ballots were a biased sample;
the results were not representative of the population. In fairness, it should be noted that the Literary
Digest used a similar master list in 1932 and predicted the popular vote within 1 percentage point.2
155
156
Chapter 8
Obviously, researchers want to avoid biased samples; random sampling seems like the
appropriate solution. The problem, however, is Step 2 on page 152. For almost all research
problems, it is impossible to identify every member of the population. What do practical
researchers do when they cannot obtain a random sample?
Research Samples
Fortunately, the problem I have posed is not a serious one. For researchers whose immediate goal
is to generalize their results directly to a population, techniques besides random sampling produce
results that mirror the population in question. Such nonrandom (though carefully controlled) samples
are used by a wide variety of people and organizations. The public opinion polls reported by Gallup,
FiveThirtyEight, and SurveyUSA are based on carefully selected nonrandom samples. Inventory
procedures in large retail organizations rely on sampling. What you know about your blood is based
on an analysis of a small sample. (Thank goodness!) And what you think about your friends and
family is based on just a sample of their behavior.
More importantly, the immediate goal of most researchers is simply to determine if differences
in the independent variable correspond to differences in the dependent variable. Research
samples, often called convenience samples, are practical and expedient. They are seldom obtained
using random sampling procedures. If researchers find differences using convenience samples,
generalization then becomes a goal.
If the researchers find a difference, they conduct the experiment again, perhaps varying the
independent or dependent variable. If several similar experiments (often called replications) produce
a coherent pattern of differences, the researchers and their colleagues conclude that the differences
are general, even though no random samples were used at any time. Most of the time, they are
correct. For these researchers, the representativeness of their samples is not of immediate concern.
PROBLEMS
8.4. Suppose a questionnaire about educational accomplishments is distributed. Do you think
that some recipients will be more likely to respond than others? Which ones? If the
sample is biased, will it overestimate or underestimate the educational accomplishments
of the population?
8.5. Consider as a population the students at State U. whose names are listed alphabetically in
the student directory. Are the following samples biased or random?
a. Every fifth name on the list
b. Every member of the junior class
c. 150 names drawn from a box that contains all the names in the directory
d. One name chosen at random from the directory
e. A random sample from those taking the required English course
2
See the Literary Digest, August 22, 1936, and November 14, 1936.
Samples, Sampling, Distributions, and Confidence Intervals
157
Sampling Distributions
A sampling distribution is always the sampling distribution of a Sampling distribution
particular statistic. Thus, there is a sampling distribution of the mean, a Theoretical distribution of a
sampling distribution of the variance, a sampling distribution of the range, statistic based on all possible
random samples drawn from the
and so forth. Here is a description of an empirical sampling distribution of same population.
a statistic.
Think about many random samples (each with the same N) all drawn from the same
population. The same statistic (for example, the mean) is calculated for each sample. All of
these statistics are arranged into a frequency distribution and graphed as a frequency polygon.
The mean and the standard deviation of the frequency distribution are calculated. Now, imagine
that the frequency polygon is a normal curve. If the distribution is normal and you have its
mean and standard deviation, you can calculate z scores and find probabilities associated with
particular sample means.
The sampling distributions that statisticians and textbooks use are theoretical rather than
empirical. Theoretical distributions, too, give you probabilities that correspond to values of
particular statistics.
Sampling distributions are so important that statisticians have special
Expected value
names for their mean and standard deviation. The mean of a sampling Mean value of a random
distribution is called the expected value and the standard deviation is called variable over an infinite number
of samplings.
the standard error. In this and other statistical contexts, the term error
means deviations or random variation. The word error was adopted in the Standard error
19th century when random variation was referred to as the Ã¢â‚¬Å“normal law of Standard deviation of a
sampling distribution.
error.Ã¢â‚¬Â Of course, error sometimes means mistake, so you will have to be
alert to the context when this word appears (or you may make an error).
To conclude, a sampling distribution is the theoretical distribution of a statistic based
on random samples that all have the same N. Sampling distributions are used by all who use
inferential statistics, regardless of the nature of their research samples. To see what different
sampling distributions look like, peek ahead to Figure 8.3, Figure 8.5, Figure 11.4, and Figure
14.1.
Sampling Distribution of the Mean
LetÃ¢â‚¬â„¢s turn to a particular sampling distribution, the sampling distribution of the mean. Remember
the population of 20 scores that you sampled from earlier in this chapter? For Problem 8.2, you
drew a sample of 10 and calculated the mean. For almost every sample in your class, the mean
was different. Each, however, was an estimate of the population mean.
My example problem in the section on random sampling used N = 8; my one sample
produced a mean of 8.50. Now, think about drawing 200 samples with N = 8 from the population
of Table 8.1 scores, calculating the 200 means, and constructing a frequency polygon. You may
already be thinking, Ã¢â‚¬Å“ThatÃ¢â‚¬â„¢s a job for a computer program.Ã¢â‚¬Â Right. The result is Figure 8.1.
158
Chapter 8
F I G U R E 8 . 1 Empirical sampling distribution of 200 means from the
population in Table 8.1. For each sample mean, N=8
The characteristics of a sampling distribution of the mean are:
1. Every sample is a random sample, drawn from a specified population.
2. The sample size (N) is the same for all samples.
3. The number of samples is very large.
4. The mean Ã°Ââ€˜â€¹ is calculated for each sample.3
5. The sample means are arranged into a frequency distribution.
I hope that when you looked at Figure 8.1, you were at least suspicious that it might be
the ubiquitous normal curve. It is. Now you are in a position that educated people often find
themselves: What you learned in the past, which was how to use the normal curve for scores
(X), can be used for a different problemÃ¢â‚¬â€describing the relationship between Ã°Ââ€˜â€¹ and Ã‚Âµ.
Of course, the normal curve is a theoretical curve, and I presented you with an empirical
curve that only appears normal. I would like to let you prove for yourself that the form of
a sampling distribution of the mean is a normal curve, but, unfortunately, that requires
mathematical sophistication beyond that assumed for this course. So I will resort to a timehonored teaching techniqueÃ¢â‚¬â€an appeal to authority.
3
To create a sampling distribution of a statistic other than the mean, substitute that statistic at this step.
Samples, Sampling, Distributions, and Confidence Intervals
159
Central Limit Theorem
The authority I appeal to is mathematical statistics, which proved a theorem
called the Central Limit Theorem:
Central Limit Theorem
The sampling distribution of
the mean approaches a normal
curve as N gets larger.
For any population of scores, regardless of form, the sampling
distribution of the mean approaches a normal distribution as N
(sample size) gets larger. Furthermore, the sampling distribution of the mean has a
mean (the expected value) equal to Ã‚Âµ and a standard deviation (the standard error)
equal to ÃÆ’Ã¢Ââ€žÃ¢Ë†Å¡N.
This appeal to authority resulted in a lot of information about the sampling distribution of
the mean. To put this information into list form:
1. The sampling distribution of the mean approaches a normal curve as N increases
2. For a population with a mean, Ã‚Âµ, and a standard deviation, ÃÆ’,
a. The mean of the sampling distribution (expected value) = Ã‚Âµ
b. The standard deviation of the sampling distribution (standard error) = Ã°ÂÅ“Å½/Ã¢Ë†Å¡ Ã°Ââ€˜Â.
Here are two additional points about terminology:
1. The symbol for the expected value of the mean is E(Ã°Ââ€˜â€¹)
2. The symbol for the standard error of the mean is Ã°ÂÅ“Å½Ã°Ââ€˜â€¹
Thus,
Ã°ÂÅ“Å½Ã°Ââ€˜â€¹ =
Ã°ÂÅ“Å½
Ã¢Ë†Å¡Ã°Ââ€˜Â
The most remarkable thing about the Central Limit Theorem is that it works regardless
of the form of the original distribution. Figure 8.2 shows two populations at the top and
three sampling distributions below each population. On the left is a rectangular distribution
of playing cards (from Figure 7.1), and on the right is the bimodal distribution of number
choices (from Figure 7.5). The three sampling distributions of the mean have Ns equal to 2, 8,
and 30. The take-home message of the Central Limit Theorem is that, regardless of the form
of a distribution, the form of the sampling distribution of the mean approaches normal if N is
large enough.
How large must N be for the sampling distribution of the mean to be normal? The
traditional answer is 30 or more. However, if the population itself is symmetrical, then sampling
distributions of the mean will be normal with Ns much smaller than 30. In contrast, if the
population is severely skewed, Ns of more than 30 will be required.
Finally, the Central Limit Theorem does not apply to all sample statistics. Sampling
distributions of the median, standard deviation, variance, and correlation coefficient are not
normal distributions. The Central Limit Theorem does apply to the mean, which is a most
important and popular statistic. (In a frequency count of all statistics, the mean is the mode.)
160
Chapter 8
F I G U R E 8 . 2 Populations of playing cards and number choices. Sampling
distributions from each population with sample sizes of N=2, N=8, and N=30
PROBLEMS
8.6. The standard deviation of a sampling distribution is called the_________and the mean of a
sampling distribution is called the_____________________________.
8.7. a. Write the steps needed to construct an empirical sampling distribution of the range.
b. What is the name of the standard deviation of this sampling distribution?
c. Think about the expected value of the range compared to the population range. Write a
statement about the relationship.
8.8. Describe the Central Limit Theorem in your own words.
Samples, Sampling, Distributions, and Confidence Intervals
8.9. In Chapter 6, you learned how to use a regression equation to predict a score (Ã…Â¶). Ã…Â¶ is a
statistic, so naturally it has a sampling distribution with its own standard error. What can
you conclude if the standard error is very small? Very large?
Calculating the Standard Error of the Mean
Calculating the standard error of the mean is fairly simple. I will illustrate with an example that
you will use again. For the population of scores in Table 8.1, ÃÆ’ is 2. For a sample size of eight,
The Effect of Sample Size on the
Standard Error of the Mean
As you can see by looking at the formula
for the standard error of the mean, Ã°ÂÅ“Å½Ã°Ââ€˜â€¹
becomes smaller as N gets larger. Figure
8.3 shows four sampling distributions of
the mean, all based on the population of
numbers in Table 8.1. The sample sizes
are 2, 4, 8, and 16. A sample mean of 10 is
included in all four figures as a reference
point. Notice that as N increases, a sample
mean of 10 becomes less and less likely.
The importance of sample size will become
more apparent as your study progresses.
F I G U R E 8 . 3 Sampling distributions of
the mean for four different sample sizes.
All samples are drawn from the population
in Table 8.1. Note how a sample mean of
10 becomes rarer and rarer as Ã°ÂÅ“Å½Ã°Ââ€˜â€¹ becomes
smaller.
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Chapter 8
Determining Probabilities About Sample Means
To summarize where we are at this point: Mathematical statisticians have produced a
mathematical invention, the sampling distribution. One particular sampling distribution, the
sampling distribution of the mean, is a normal curve, they tell us. Fortunately, having worked
problems in Chapter 7 about normally distributed scores, we are in a position to check this
claim about normally distributed means.
One check is fairly straightforward, given that I already have the 200 sample means from
the population in Table 8.1. To make this check, I will determine the proportion of sample
means that are above a specified point, using the theoretical normal curve (Table C in Appendix
C). I can then compare this theoretical proportion to the proportion of the 200 sample means
that are actually above the specified point. If the two numbers are similar, I have evidence that
the normal curve can be used to answer questions about sample means.
The z score for a sample mean drawn from a sampling distribution with mean Ã‚Âµ and
standard error Ã°ÂÅ“Å½/Ã¢Ë†Å¡ Ã°Ââ€˜Â is
z=
Ã°Ââ€˜â€¹-Ã°ÂÅ“â€¡
Ã°ÂÅ“Å½Ã°Ââ€˜â€¹
Any sample mean will do for this comparison; I will use 10.0. The mean of the population
is 9.0 and the standard error (for N = 8) is 0.707. Thus,
z=
Ã°Ââ€˜â€¹-Ã‚Âµ
Ã°ÂÅ“Å½Ã°Ââ€˜â€¹
=
10.0 - 9.0
= 1.41
0.707
By consulting Table C, I see that the proportion of the curve above a z value of 1.41 is
.0793. Figure 8.4 shows the sampling distribution of the mean for N = 8. Sample means are on
the horizontal axis; the .0793 area beyond Ã°Ââ€˜â€¹ = 10 is shaded.
F I G U R E 8 . 4 Theoretical sampling distribution of the mean from the
population in Table 8.1. For each sample, N = 8
Samples, Sampling, Distributions, and Confidence Intervals
How accurate is this theoretical prediction of .0793? When I looked at the distribution of
sample means that I used to construct Figure 8.1, I found that 13 of the 200 had means of 10
or more, a proportion of .0650. Thus, the theoretical prediction is off by less than 1Ã‚Â½%. ThatÃ¢â‚¬â„¢s
not bad; the normal curve model passes the test. (The validity of the normal curve model for
the sampling distribution of the mean has been established with a mathematical proof as well.)
PROBLEMS
*8.10. When the population parameters are known, the standard error of the mean is
ÃÆ’Ã°Ââ€˜â€¹ = Ã°ÂÅ“Å½/Ã¢Ë†Å¡ Ã°Ââ€˜Â. The following table gives four ÃÆ’ values and four N values. For each
combination, calculate ÃÆ’Ã°Ââ€˜â€¹ and enter it in the table.
N
1
ÃÆ’
2
4
8
1
4
16
64
8.11. On the basis of the table you constructed in Problem 8.10, write a precise verbal statement
about the relationship between ÃÆ’Ã°Ââ€˜â€¹ and N.
8.12. To reduce ÃÆ’Ã°Ââ€˜â€¹ to one-fourth its size, you must increase N by how much?
8.13. For the population in Table 8.1, and for samples with N = 8, what proportion of the sample
means will be 8.5 or less?
8.14. For the population in Table 8.1, and for samples with N = 16, what proportion of the
sample means will be 8.5 or less? 10 or greater?
8.15. As you know from the previous chapter, for IQs, Ã‚Âµ = 100 and ÃÆ’ = 15. What is the
probability that a first-grade classroom of 25 students who are chosen randomly from the
population will have a mean IQ of 105 or greater? 90 or less?
8.16. Now you are in a position to return to the story of the two students at the beginning of
this chapter. Find, for the junior, the probability that a sample of 40 from a population
with Ã‚Âµ = $95,800 and ÃÆ’ = $25,000 would produce a mean of $91,900 or less. Write an
interpretation.
Based on your answer to Problem 8.16, the junior might say to the sophomore,
Ã¢â‚¬Å“ . . . and so, the mean of $91,900 isnÃ¢â‚¬â„¢t a trustworthy measure of the population.
The standard deviation is large, and with N = 40, samples can bounce all over the
place. Why, if the actual mean for our campus is the same as the national average of
$82,000, we would expect about one-sixth of all random samples of 40 students to
have means of $91,900 or less.Ã¢â‚¬Â
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Chapter 8
Ã¢â‚¬Å“Yeah,Ã¢â‚¬Â said the sophomore, Ã¢â‚¬Å“but that is a big IF. What if the campus mean is
$91,900? Then the sample we got was right on the nose. After all, a sample mean is
an unbiased estimator of the population parameter.Ã¢â‚¬Â
Ã¢â‚¬Å“I see your point. And I see mine, too. It seems like either of us could be correct.
That leaves me uncertain about the real campus parameter.Ã¢â‚¬Â
Ã¢â‚¬Å“Me, too. LetÃ¢â‚¬â„¢s go get another ice cream cone.Ã¢â‚¬Â
Not all statistical stories end with so much uncertainty (or calories). However, I said
that one of the advantages of random sampling is that you can measure the uncertainty. You
measured it, and there was a lot. Remember, if you agree to use a sample, you agree to accept
some uncertainty about the results.
Constructing a Sampling Distribution When ÃÆ’ Is Not Available
LetÃ¢â‚¬â„¢s review what you just did. You answered some questions about sample means by
relying on a table of the normal curve. Your justification for saying that sample means are
distributed normally was the Central Limit Theorem. The Central Limit Theorem always applies
when the sample size is adequate and you know ÃÆ’, both of which were true for the problems
you worked. For those problems, you were given ÃÆ’ or calculated it from the population data
you had available.
In the world of empirical research, however, you often do not know ÃÆ’ and you donÃ¢â‚¬â„¢t
have population data to calculate it. Because researchers are always inventing new dependent
variables, an unknown ÃÆ’ is common. Without ÃÆ’, the justification for using the normal curve
evaporates. What to do if you donÃ¢â‚¬â„¢t have ÃÆ’? Can you suggest something?
One solution is to use Ã…Â as an estimate of ÃÆ’. (Was that your suggestion?) This was the
solution used by researchers about a century ago. They knew that Ã…Â was only an estimate and
that the larger the sample, the better the estimate. Thus, they chose problems for which they
could gather huge samples. (Remember Karl Pearson and Alice LeeÃ¢â‚¬â„¢s data on fatherÃ¢â‚¬â€œdaughter
heights? They had a sample of 1376.) Very large samples produce an Ã…Â that is identical to ÃÆ’, for
all practical purposes.
Other researchers, however, could not gather that much data. One of those we remember
today is W. S. Gosset (1876Ã¢â‚¬â€œ1937), who worked for Arthur Guinness, Son & Co., a brewery
headquartered in Dublin, Ireland. Gosset had majored in chemistry and mathematics at Oxford,
and his job at Guinness was to make recommendations to the brewers that were based on
scientific experiments. The experiments, of course, used samples.
Gosset was familiar with the normal curve and the strategy of using large samples
to accurately estimate ÃÆ’. Unfortunately, though, his samples were small. Gosset (and other
statisticians) knew that such small-sample Ã…Â values were not accurate estimators of ÃÆ’ and thus
the normal curve could not be relied on for accurate probabilities.
GossetÃ¢â‚¬â„¢s solution was to work out a new set of distributions based on Ã…Â rather than on ÃÆ’. He
found that the distribution depended on the sample size, with a different distribution for each N.
Samples, Sampling, Distributions, and Confidence Intervals
These distributions make up a family of curves that have come to be called the t distribution.4
The t distribution is an important tool for those who analyze data. I will use it for confidence
interval problems in this chapter and for four other kinds of problems in later chapters.
The t Distribution
The different curves that make up the t distribution are distinguished from Degrees of freedom
one another by their degrees of freedom. Degrees of freedom (abbreviated Concept in mathematical
df) range from 1 to Ã¢Ë†Å¾. Knowing the degrees of freedom for your data tells statistics that determines the
distribution that is appropriate
you which t distribution to use.5 Determining the correct number of degrees for sample data.
of freedom for a particular problem can become fairly complex. For the
problems in this chapter, however, the formula is simple: df = N Ã¢â‚¬â€œ 1. Thus, if
the sample consists of 12 members, df = 11. In later chapters, I will give you a more thorough
explanation of degrees of freedom (and additional formulas).
Figure 8.5 is a picture of three t distributions. Their degrees of freedom are 2, 9, and Ã¢Ë†Å¾.
You can see that as the degrees of freedom increase, less and less of the curve is in the tails.
Note that the t values on the horizontal axis are quite similar to the z scores used with the normal
curve.
F I G U R E 8 . 5 Three different t distributions
Gosset spent several months in 1906Ã¢â‚¬â€œ07 studying with Karl Pearson in London. It was during this period that the t
distribution was developed.
5
Traditionally, the t distribution is written with a lowercase t. A capital T is used for another distribution (covered in Chapter 15) and for standardized test scores. However, because some early computer programs did not print lowercase letters, t
became T on some printouts (and often in text based on that printout). Be alert.
4
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Chapter 8
The t Distribution Table
Look at Table D (p. 388), the t distribution table in Appendix C. The first column shows degrees
of freedom, ranging from 1 to Ã¢Ë†Å¾. Degrees of freedom are determined by an analysis of the
problem that is to be solved.
There are three rows across the top of the table; the row you use depends on the kind of
problem you are working on. In this chapter, use the top row because the problems are about
confidence intervals. The second and third rows are used for problems in Chapter 9 and Chapter
10.
The body of Table D contains t values. They are used most frequently to find a probability
or percent associated with a particular t value. For data with a df value intermediate between
tabled values, be conservative and use the t value associated with the smaller df.
Table D differs in s...
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