Campbellsville University Statistics Normal Distribution Discussion

  

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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Answer Key: Jill Schmidlkofer
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Photo Credits – Chapter 1
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Ronald A. Fisher – R.A. Fisher portrait, 0006973, Special Collections Research Center, North Carolina State
University Libraries, Raleigh, North Carolina
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Printed in the United States of America by Walsworth ®
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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. Spatz’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, “Statistics”? 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
ŝ as an Estimate of σ 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 σ 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 Reinforcement—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 × 3 Design 291 Comparing Levels Within a Factor—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 × 2 Table 310 Effect Size Indexes for 2 × 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 conversation—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. – 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 book’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: • Data sets are approached with an attitude of exploration. • Changes in statistical practice over the years are acknowledged, especially the recent emphasis on effect sizes and confidence intervals. • Criticism of null hypothesis significance testing (NHST) is explained. • Examples and problems represent a variety of disciplines and everyday life. • Most problems are based on actual studies rather than fabricated scenarios. • Interpretation is emphasized throughout. • Problems are interspersed within a chapter, not grouped at the end. • Answers to all problems are included. • Answers are comprehensively explained—over 50 pages of detail. • A final chapter, Choosing Tests and Writing Interpretations, requires active responses to comprehensive questions. Preface • Effect size indexes are treated as important descriptive statistics, not add-ons to NHST. • Important words and phrases are defined in the margin when they first occur. • Objectives, which open each chapter, serve first for orientation and later as review items. • Key Terms are identified for each chapter. • Clues to the Future alert students to concepts that come up again. • Error Detection boxes tell ways to detect mistakes or prevent them. • Transition Passages alert students to a change in focus in chapters that follow. • Comprehensive Problems encompass all (or most) of the techniques in a chapter. • 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: • Solve statistical problems • Understand and explain statistical reasoning • Choose appropriate statistical techniques for common research designs • 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 book’s website, the text’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. I’m particularly grateful to Grace Oxley for a cover design that conveys exploration, and to Liann Lech, who copyedited for clarity and consistency. Walsworth® turned a messy collection of files into a handsome book—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 I’ve always had a touch of the teacher in me—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 book’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. I’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 “successes” or “failures.” 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 “head” 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 2015–2016. 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 here’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 you’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, I’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 “chances in a hundred.” 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 “success”). 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, you’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. Here’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 “odds of 5:1” 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 let’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 “best estimate” 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 valuable—the normal distribution. With contributions from Abraham de Moivre (1667– 1754) and Pierre-Simon Laplace (1749–1827), Carl Friedrich Gauss (1777– 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 carpenter’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) (1796– 1874), a Belgian who showed that many social and biological measurements are distributed normally. Quetelet, who knew about the “law of error” 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 today’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 “the founder of the most important science in the world” (Cook, 1913, p. 238 as quoted in Maindonald & Richardson, 2004). Quetelet’s work also gave Francis Galton the idea that characteristics we label “genius” 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 “should” 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 𝑋 = a raw score µ = the mean of the distribution σ = 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). 4 133 134 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 score—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 area—50%—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 –3σ to +3σ. (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 –4σ to +4σ.) concave downward arc, or vice Another point about the normal distribution is that the two inflection versa. points in the curve are at exactly –1σ and +1σ. The inflection points are where the curve is the steepest—that is, where the curve changes from bending upward to bending over. (See the points above –1σ and +1σ in Figure 7.4 and think of walking up, over, and down a bell-shaped hill.) To end this introductory section, here’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 isn’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 isn’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 (∞) (column C) F I G U R E 7 . 6 The normal distribution showing the probabilities of certain z scores 135 136 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 = –1.00. Thus, all the proportions in Figure 7.6 were derived by finding proportions associated with z = 1.00. Don’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 = –2.01 7.9. Is the distribution in Figure 7.5 theoretical or empirical? As I’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 “large” 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. 137 138 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 –1.33. Questions of “How Many?” You can answer questions of “How many?” 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? Here’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 don’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 “top 10%” 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 “area beyond z” 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= X–µ σ Multiplying both sides by σ produces (z)(σ) = 𝑋 – µ Adding µ 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 𝑋 = µ + (z)(σ) 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 µ = 58 and σ = 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 141 142 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 –0.43. (This time you are dealing with a z score below the mean, where all z scores are negative.) Applying z = –0.43, you get 𝑋 = µ + (z)(σ) = 58 + (–0.43)(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 “higher than 98% of the population.” What IQ is required to qualify? 7.16. The mean height of American women aged 20–29 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 20–29 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 5’2”? 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 “average” 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 = –0.67. Therefore, (2)(.2486) = .4972 or 49.72%. So approximately 50% of the population is classified as “average,” using the “IQ = 90 to 110” definition. “Average” 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. I’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 15 143 144 Chapter 7 The area beyond z = –0.67 is .2514 and the area beyond z = –2.00 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 𝜇 = 35 and 𝜎 = 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 2½% and the lower 2½% receive attention. Turn to Table C and find the z score that separates the extreme 2½% of the curve from the rest. (Of course, the z score associated with the lowest 2½% 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 𝑋 = µ + (z)(σ). Using the values given, plus the z score you memorized, you get the following: Upper Score 𝑋 = µ + (z)(σ) = 71 + (1.96)(9) = 71 + 17.6 = 88.6 or 89 bpm Lower Score 𝑋 = µ – (z)(σ) = 71 – (1.96)(9) = 71 – 17.6 = 53.4 or 53 bpm What do you make of these statistics? (Look and respond; my informal response follows.) “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. I’m surprised.” 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 “extreme 5%” 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 𝜇 = 13.68 inches and 𝜎 = 4.83 inches. How many harvestable trees would be expected from 100 acres? 145 146 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 I’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, “The Unicorn, the Normal Curve, and Other Improbable Creatures.” Caution is always in order when you are using theoretical distributions to make predictions about empirical events. However, don’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 6’8”. 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: ∑X = 3600 ∑X 2 = 130,000 Assume that the heights of hobbits are normally distributed. Find µ and σ 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 σ 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, “Tales of Distributions,” 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 I’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, I’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. “Well, the average family income for college students is $95,800, nationwide,” said the junior, looking up from a book (Eagen et al., 2018). “I’m sure the mean for this campus is at least that much.” “I don’t think so,” the sophomore replied. “I know lots of students who have only their own resources or come from pretty poor families. I’ll bet you a dollar the mean for students here at State U. is below the national average.” “You’re on,” 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. “I told you so; the mean here is almost $10,000 less than the national average.” Disappointed, the junior immediately began to review their procedures. Rather quickly, a light went on. “Actually, this average of $86,000 is meaningless—here’s why. Those 10 students aren’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.” 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. “Pay,” demanded the sophomore. “OK, OK, . . . here!” 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. “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?” “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 don’t I just apply my winnings toward a couple of ice cream cones.” 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 µ = 9 and σ = 2. Using these 20 scores allows us to satisfy Requirements 1 and 2. As for Requirement 3, I’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; µ = 9, σ = 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. I’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 15–19 (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 doesn’t match a score in the population, but at least you are started. From this point, you can read two-digit numbers in any direction—up, down, or sideways—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 𝑋. 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 “all telephone books in the U.S., rosters of clubs, lists of registered voters,” 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 Roosevelt’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 “normal law of Standard deviation of a sampling distribution. error.” 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 Let’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, “That’s a job for a computer program.” 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 𝑋 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 problem—describing the relationship between 𝑋 and µ. 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 technique—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 µ and a standard deviation (the standard error) equal to σ⁄√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, µ, and a standard deviation, σ, a. The mean of the sampling distribution (expected value) = µ b. The standard deviation of the sampling distribution (standard error) = 𝜎/√ 𝑁. Here are two additional points about terminology: 1. The symbol for the expected value of the mean is E(𝑋) 2. The symbol for the standard error of the mean is 𝜎𝑋 Thus, 𝜎𝑋 = 𝜎 √𝑁 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 (Ŷ). Ŷ 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, σ 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, 𝜎𝑋 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 𝜎𝑋 becomes smaller. 161 162 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 µ and standard error 𝜎/√ 𝑁 is z= 𝑋-𝜇 𝜎𝑋 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= 𝑋-µ 𝜎𝑋 = 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 𝑋 = 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 1½%. That’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 σ𝑋 = 𝜎/√ 𝑁. The following table gives four σ values and four N values. For each combination, calculate σ𝑋 and enter it in the table. N 1 σ 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 σ𝑋 and N. 8.12. To reduce σ𝑋 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, µ = 100 and σ = 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 µ = $95,800 and σ = $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, “ . . . and so, the mean of $91,900 isn’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.” 163 164 Chapter 8 “Yeah,” said the sophomore, “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.” “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.” “Me, too. Let’s go get another ice cream cone.” 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 σ Is Not Available Let’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 σ, both of which were true for the problems you worked. For those problems, you were given σ or calculated it from the population data you had available. In the world of empirical research, however, you often do not know σ and you don’t have population data to calculate it. Because researchers are always inventing new dependent variables, an unknown σ is common. Without σ, the justification for using the normal curve evaporates. What to do if you don’t have σ? Can you suggest something? One solution is to use ŝ as an estimate of σ. (Was that your suggestion?) This was the solution used by researchers about a century ago. They knew that ŝ 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 Lee’s data on father–daughter heights? They had a sample of 1376.) Very large samples produce an ŝ that is identical to σ, for all practical purposes. Other researchers, however, could not gather that much data. One of those we remember today is W. S. Gosset (1876–1937), 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 σ. Unfortunately, though, his samples were small. Gosset (and other statisticians) knew that such small-sample ŝ values were not accurate estimators of σ and thus the normal curve could not be relied on for accurate probabilities. Gosset’s solution was to work out a new set of distributions based on ŝ rather than on σ. 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 ∞. 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 – 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 ∞. 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 1906–07 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. 4 165 166 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 ∞. 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... Purchase answer to see full attachment