StatsCloud Homework Problem?

trong>Questions –When the Referee Sees Red…

1. Which data are more variable – the age of the referees or their years of experience?How do you know?

2.What is the IV in this study?What is the DV?

3.In this study, why is a dependent t-test more appropriate than an independent t-test to analyze the data?Make sure your answer is specific with respect to what is being compared.

4.Was the number of points awarded to the competitor wearing red protective gear significantly higher than those awarded to the competitor wearing blue protective gear?How do you know, statistically speaking?

5.Was the number of points awarded to a blue competitor who was digitally transformed into a red competitor significantly higher?How do you know, statistically speaking?

6.For which comparison was the effect size the largest?How would the value be interpreted?

7.What one change or addition would you make to this study? (can’t say “none”!)

PS YC HOLOGICA L SC IENCE
Short Report
When the Referee Sees Red . . .
Norbert Hagemann, Bernd Strauss, and Jan Leiing
Westfälische Wilhelms-Universität Münster
Hill and Barton (2005) showed that wearing red sports attire has
a positive impact on one’s outcome in a combat sport (e.g., tae
kwon do or wrestling). They suggested that this effect is due to an
evolutionary or cultural association of the color red with dominance and aggression, proposing that this association triggers
a psychological effect in an athlete who wears red (or in his or
her opponent; e.g., Cuthill, Hunt, Cleary, & Clark, 1997;
Milinski & Bakker, 1990; Setchell & Wickings, 2005). Rowe,
Harris, and Roberts (2005) criticized this argument and instead
attributed the bias evident in these and other data (judo) to
differences in opponents’ visibility.
We disagree with both interpretations (see also Barton & Hill,
2005), arguing that this phenomenon is actually due to a perceptual bias in the referee. That is, we propose that the perception of colors triggers a psychological effect in referees that
can lead to bias in evaluating identical performances. Referees
and umpires exert a major influence on the outcome of sports
competitions (Plessner & Haar, 2006). Athletes frequently make
very rapid movements, and referees have to view sports competitions from a very disadvantageous perspective, so it is extremely difficult for them to make objective judgments (Oudejans et al., 2000). As a result, their judgments may show biases
like those found in other social judgments (Frank & Gilovich,
1988; Plessner & Haar, 2006; Ste-Marie & Valiquette, 1996).
Therefore, we believe that it is the referees who are the actual
cause of the advantage competitors have when they wear red.
Because the effect of red clothing on performance and on the
decisions of referees may well have been confounded in the
original data, we conducted a new experiment and found that
referees assign more points to tae kwon do competitors dressed
in red than to those dressed in blue, even when the performance
of the competitors is identical.
referee 5 8.02 years, SD 5 6.27) individually watched videotaped excerpts from sparring rounds of five different male
competitors of similar abilities. Each of two blocks contained 11
clips, with an average length of 4.4 s. The video images measured 1,024  768 pixels and were displayed on a notebook
computer with a 15.4-in. screen.
In each video, one competitor was wearing red protective gear,
and the other was wearing blue protective gear. (Underneath this
gear, each competitor wore a white tae kwon do uniform.) The two
blocks contained the same clips, but with the colors of the competitors reversed. We reversed the colors using digital graphics,
animation, and image-compositing software (Adobe After Effects
7.0).
After viewing each clip, participants indicated how many
points they would award the red and the blue competitors.
Following the rules of the World Taekwondo Federation,1 participants awarded points when permitted techniques were used
to deliver attacks to the legal scoring areas of the body: Specifically, 1 point was awarded for an attack to the trunk protector
(fist and foot techniques), and 2 points were awarded for an attack to the face (only attacks by foot technique are permitted).
Additional points could be awarded if a contestant knocked
down his opponent. Prohibited acts could be counted as a deduction of 1 point.
The video clips were presented in random order within each
block, and the order of the blocks was counterbalanced across
participants. For each referee, we calculated the total number of
points for the red and blue competitors, and these values were
subjected to separate dependent t tests. We used Cohen’s d as
our measure of effect size. We expected that changing the color
of the protective gear from blue to red would lead to an increase
in points awarded, whereas changing the color from red to blue
would have the opposite effect.
METHOD
We investigated the effect of the color of the protective gear
(trunk and head protectors) in tae kwon do on the decisions of
referees. A total of 42 experienced referees (13 female, 29 male;
mean age 5 29.31 years, SD 5 10.56; mean experience as a
Address correspondence to Norbert Hagemann, Department of Sport
Psychology, University of Münster, Horstmarer Landweg 62b, 48149
Münster, Germany, e-mail: nhageman@uni-muenster.de.
Volume 19—Number 8
RESULTS AND DISCUSSION
Figure 1 shows the effect of the color of the protective gear. The
competitor wearing red protective gear was awarded an average
1
In a regular competition, the match score is the sum of points in three 2-min
rounds. Unless there is a knockout, withdrawal, or disqualification, the winner
is usually determined by points: The winner has the higher final score, exceeds
the opponent’s score by 7 points, or reaches the maximum of 12 points.
Copyright r 2008 Association for Psychological Science
769
Seeing Red
10
9
8
Mean Points
7
6
5
4
3
Competitors A Competitors B
(red)
(blue)
Competitors A Competitors B
(blue)
(red)
Original
Color-Reversed
2
1
0
Version
Fig. 1. Mean number of points awarded to tae kwon do competitors in the original and color-reversed versions of the video
clips. Each clip depicted a sparring round with one competitor dressed in red and the other dressed in blue. Competitors A
wore red in the original clips and blue in the color-reversed clips, and Competitors B wore blue in the original clips and red
in the color-reversed clips. Error bars indicate standard errors.
of 13% (0.94 points) more points than the competitor wearing
blue protective gear, t(41) 5 2.85, p < .01, d 5 0.35. The number of points awarded increased for a blue competitor who was digitally transformed into a red competitor, t(41) 5 2.45, p < .01 (one-tailed), d 5 0.36, and decreased for a red competitor who was digitally transformed into a blue competitor, t(41) 5 1.66, p < .05 (one-tailed), d 5 0.25. The gender of the referee, total number of points awarded in the two versions of the video clip (original vs. color-reversed), and the order in which the two versions were presented had no effect on the referees’ decisions. Thus, competitors dressed in red are awarded more points than competitors dressed in blue, even when their performance is identical. The effect found in this experiment can also explain why the effect of clothing color on the outcome is stronger when competitors have similar abilities than when there is a large asymmetry in their abilities (Hill & Barton, 2005). Referees’ decisions will ‘‘tip the scales’’ when athletes are relatively well matched, but have relatively small influence when one is clearly superior. Even though the color of athletes’ sportswear may well exert an influence on their performance (through associations with dominance or differences in visibility of the opponent), we argue that the referees are responsible for the advantage conveyed to athletes who wear red. Although there is a need for further research (including research on the effects of different 770 colors), our results suggest a need to change the rules (i.e., forbid red sports attire) or support referees by providing electronic decision-making aids (e.g., electronic trunk protectors) in those sports in which this color bias may be a problem. REFERENCES Barton, R.A., & Hill, R.A. (2005). Sporting contests: Seeing red? Putting sportswear in context (reply). Nature, 437, E10–E11. Cuthill, I.C., Hunt, S., Cleary, C., & Clark, C. (1997). Colour bands, dominance, and body mass regulation in male zebra finches (Taeniopygia guttata). Proceedings of the Royal Society B: Biological Sciences, 264, 1093–1099. Frank, M.G., & Gilovich, T. (1988). The dark side of self- and social perception: Black uniforms and aggression in professional sports. Journal of Personality and Social Psychology, 54, 74–85. Hill, R.A., & Barton, R.A. (2005). Red enhances human performance in contests. Nature, 435, 293. Milinski, M., & Bakker, T.C.M. (1990). Female sticklebacks use male coloration in mate choice and hence avoid parasitized males. Nature, 344, 330–333. Oudejans, R.R., Verheijen, R., Bakker, F.C., Gerrits, J.C., Steinbruckner, M., & Beek, P.J. (2000). Errors in judging ‘offside’ in football. Nature, 404, 33. Plessner, H., & Haar, T. (2006). Sports performance judgments from a social cognitive perspective. Psychology of Sport and Exercise, 7, 555–575. Volume 19—Number 8 Norbert Hagemann, Bernd Strauss, and Jan Leiing Rowe, C., Harris, J.M., & Roberts, S.C. (2005). Sporting contests: Seeing red? Putting sportswear in context. Nature, 437, E10. Setchell, J.M., & Wickings, E.J. (2005). Dominance, status signals and coloration in male mandrills (Mandrillus sphinx). Ethology, 111, 25–50. Volume 19—Number 8 Ste-Marie, D.M., & Valiquette, S.M. (1996). Enduring memory-influenced biases in gymnastic judging. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 1498–1502. (RECEIVED 12/16/07; REVISION ACCEPTED 2/13/08) 771 Ed iti o n © Deborah Batt 10 Statistics for the Behavioral Sciences Frederick J Gravetter The College at Brockport, State University of New York Larry B. WaLLnau The College at Brockport, State University of New York Australia ● Brazil ● Mexico ● Singapore ● United Kingdom ● United States This is an electronic version of the print textbook. 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Find your local representative at www.cengage.com Cengage Learning products are represented in Canada by nelson Education, Ltd. to learn more about Cengage Learning Solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Printed in Canada Print Number: 01 Print Year: 2015 B RiEF Co n tEn t S CHAPtER 1 Introduction to Statistics 1 CHAPtER 2 Frequency Distributions CHAPtER 3 Central Tendency CHAPtER 4 Variability CHAPtER 5 z-Scores: Location of Scores and Standardized Distributions 131 CHAPtER 6 Probability CHAPtER 7 Probability and Samples: The Distribution of Sample Means 193 CHAPtER 8 Introduction to Hypothesis Testing 223 CHAPtER 9 Introduction to the t Statistic C H A P t E R 10 CHAPtER 11 33 67 99 159 267 The t Test for Two Independent Samples 299 The t Test for Two Related Samples 335 C H A P t E R 12 Introduction to Analysis of Variance 365 C H A P t E R 13 Repeated-Measures Analysis of Variance 413 C H A P t E R 14 Two-Factor Analysis of Variance (Independent Measures) 447 CHAPtER 15 Correlation 485 C H A P t E R 16 Introduction to Regression 529 C H A P t E R 17 The Chi-Square Statistic: Tests for Goodness of Fit and Independence 559 C H A P t E R 18 The Binomial Test 603 iii Co n tEn t S CHAPtER 1 Introduction to Statistics PREVIEW 1 2 1.1 Statistics, Science, and Observations 2 1.2 Data Structures, Research Methods, and Statistics 10 1.3 Variables and Measurement 18 1.4 Statistical Notation 25 Summary 29 Focus on Problem Solving 30 Demonstration 1.1 30 Problems 31 CHAPtER 2 Frequency Distributions PREVIEW 33 34 2.1 Frequency Distributions and Frequency Distribution Tables 35 2.2 Grouped Frequency Distribution Tables 38 2.3 Frequency Distribution Graphs 42 2.4 Percentiles, Percentile Ranks, and Interpolation 49 2.5 Stem and Leaf Displays 56 Summary 58 Focus on Problem Solving 59 Demonstration 2.1 60 Demonstration 2.2 61 Problems 62 v vi CONTENTS CHAPtER 3 Central Tendency PREVIEW 67 68 3.1 Overview 68 3.2 The Mean 70 3.3 The Median 79 3.4 The Mode 83 3.5 Selecting a Measure of Central Tendency 86 3.6 Central Tendency and the Shape of the Distribution 92 Summary 94 Focus on Problem Solving 95 Demonstration 3.1 96 Problems 96 CHAPtER 4 Variability PREVIEW 99 100 4.1 Introduction to Variability 101 4.2 Defining Standard Deviation and Variance 103 4.3 Measuring Variance and Standard Deviation for a Population 108 4.4 Measuring Standard Deviation and Variance for a Sample 111 4.5 Sample Variance as an Unbiased Statistic 117 4.6 More about Variance and Standard Deviation 119 Summary 125 Focus on Problem Solving 127 Demonstration 4.1 128 Problems 128 z-Scores: Location of Scores C H A P t E R 5 and Standardized Distributions PREVIEW 132 5.1 Introduction to z-Scores 133 5.2 z-Scores and Locations in a Distribution 135 5.3 Other Relationships Between z, X, 𝛍, and 𝛔 138 131 CONTENTS vii 5.4 Using z-Scores to Standardize a Distribution 141 5.5 Other Standardized Distributions Based on z-Scores 145 5.6 Computing z-Scores for Samples 148 5.7 Looking Ahead to Inferential Statistics 150 Summary 153 Focus on Problem Solving 154 Demonstration 5.1 155 Demonstration 5.2 155 Problems 156 CHAPtER 6 Probability PREVIEW 159 160 6.1 Introduction to Probability 160 6.2 Probability and the Normal Distribution 165 6.3 Probabilities and Proportions for Scores from a Normal Distribution 172 6.4 Probability and the Binomial Distribution 179 6.5 Looking Ahead to Inferential Statistics 184 Summary 186 Focus on Problem Solving 187 Demonstration 6.1 188 Demonstration 6.2 188 Problems 189 Probability and Samples: The Distribution C H A P t E R 7 of Sample Means PREVIEW 194 7.1 Samples, Populations, and the Distribution of Sample Means 194 7.2 The Distribution of Sample Means for any Population and any Sample Size 199 7.3 Probability and the Distribution of Sample Means 206 7.4 More about Standard Error 210 7.5 Looking Ahead to Inferential Statistics 215 193 viii CONTENTS Summary 219 Focus on Problem Solving 219 Demonstration 7.1 220 Problems 221 CHAPtER 8 Introduction to Hypothesis Testing PREVIEW 223 224 8.1 The Logic of Hypothesis Testing 225 8.2 Uncertainty and Errors in Hypothesis Testing 236 8.3 More about Hypothesis Tests 240 8.4 Directional (One-Tailed) Hypothesis Tests 245 8.5 Concerns about Hypothesis Testing: Measuring Effect Size 250 8.6 Statistical Power 254 Summary 260 Focus on Problem Solving 261 Demonstration 8.1 262 Demonstration 8.2 263 Problems 263 CHAPtER 9 Introduction to the t Statistic PREVIEW 268 9.1 The t Statistic: An Alternative to z 268 9.2 Hypothesis Tests with the t Statistic 274 9.3 Measuring Effect Size for the t Statistic 279 9.4 Directional Hypotheses and One-Tailed Tests 288 Summary 291 Focus on Problem Solving 293 Demonstration 9.1 293 Demonstration 9.2 294 Problems 295 267 CONTENTS CHAPtER 10 The t Test for Two Independent Samples PREVIEW ix 299 300 10.1 Introduction to the Independent-Measures Design 300 10.2 The Null Hypothesis and the Independent-Measures t Statistic 302 10.3 Hypothesis Tests with the Independent-Measures t Statistic 310 10.4 Effect Size and Confidence Intervals for the Independent-Measures t 316 10.5 The Role of Sample Variance and Sample Size in the Independent-Measures t Test Summary 322 325 Focus on Problem Solving 327 Demonstration 10.1 328 Demonstration 10.2 329 Problems 329 CHAPtER 11 The t Test for Two Related Samples PREVIEW 335 336 11.1 Introduction to Repeated-Measures Designs 336 11.2 The t Statistic for a Repeated-Measures Research Design 339 11.3 Hypothesis Tests for the Repeated-Measures Design 343 11.4 Effect Size and Confidence Intervals for the Repeated-Measures t 347 11.5 Comparing Repeated- and Independent-Measures Designs 352 Summary 355 Focus on Problem Solving 358 Demonstration 11.1 358 Demonstration 11.2 359 Problems 360 CHAPtER 12 Introduction to Analysis of Variance PREVIEW 366 12.1 Introduction (An Overview of Analysis of Variance) 366 12.2 The Logic of Analysis of Variance 372 12.3 ANOVA Notation and Formulas 375 365 x CONTENTS 12.4 Examples of Hypothesis Testing and Effect Size with ANOVA 383 12.5 Post Hoc Tests 393 12.6 More about ANOVA 397 Summary 403 Focus on Problem Solving 406 Demonstration 12.1 406 Demonstration 12.2 408 Problems 408 CHAPtER 13 Repeated-Measures Analysis of Variance PREVIEW 413 414 13.1 Overview of the Repeated-Measures ANOVA 415 13.2 Hypothesis Testing and Effect Size with the Repeated-Measures ANOVA 420 13.3 More about the Repeated-Measures Design 429 Summary 436 Focus on Problem Solving 438 Demonstration 13.1 439 Demonstration 13.2 440 Problems 441 Two-Factor Analysis of Variance C H A P t E R 14 (Independent Measures) PREVIEW 447 448 14.1 An Overview of the Two-Factor, Independent-Measures, ANOVA: Main Effects and Interactions 448 14.2 An Example of the Two-Factor ANOVA and Effect Size 458 14.3 More about the Two-Factor ANOVA 467 Summary 473 Focus on Problem Solving 475 Demonstration 14.1 476 Demonstration 14.2 478 Problems 479 CONTENTS CHAPtER 15 Correlation PREVIEW xi 485 486 15.1 Introduction 487 15.2 The Pearson Correlation 489 15.3 Using and Interpreting the Pearson Correlation 495 15.4 Hypothesis Tests with the Pearson Correlation 506 15.5 Alternatives to the Pearson Correlation 510 Summary 520 Focus on Problem Solving 522 Demonstration 15.1 523 Problems 524 CHAPtER 16 Introduction to Regression PREVIEW 529 530 16.1 Introduction to Linear Equations and Regression 530 16.2 The Standard Error of Estimate and Analysis of Regression: The Significance of the Regression Equation 538 16.3 Introduction to Multiple Regression with Two Predictor Variables 544 Summary 552 Linear and Multiple Regression 554 Focus on Problem Solving 554 Demonstration 16.1 555 Problems 556 The Chi-Square Statistic: Tests for Goodness C H A P t E R 17 of Fit and Independence PREVIEW 560 17.1 Introduction to Chi-Square: The Test for Goodness of Fit 561 17.2 An Example of the Chi-Square Test for Goodness of Fit 567 17.3 The Chi-Square Test for Independence 573 17.4 Effect Size and Assumptions for the Chi-Square Tests 17.5 Special Applications of the Chi-Square Tests 587 582 559 xii CONTENTS Summary 591 Focus on Problem Solving 595 Demonstration 17.1 595 Demonstration 17.2 597 Problems 597 CHAPtER 18 The Binomial Test PREVIEW 603 604 18.1 Introduction to the Binomial Test 604 18.2 An Example of the Binomial Test 608 18.3 More about the Binomial Test: Relationship with Chi-Square and the Sign Test Summary 612 617 Focus on Problem Solving 619 Demonstration 18.1 619 Problems 620 A PPE N D IX E S A Basic Mathematics Review 625 A.1 A.2 A.3 A.4 A.5 Symbols and Notation 627 Proportions: Fractions, Decimals, and Percentages 629 Negative Numbers 635 Basic Algebra: Solving Equations 637 Exponents and Square Roots 640 B Statistical Tables 647 C Solutions for Odd-Numbered Problems in the Text 663 D General Instructions for Using SPSS 683 E Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests 687 Statistics Organizer: Finding the Right Statistics for Your Data References 717 Name Index 723 Subject Index 725 701 PREFACE M any students in the behavioral sciences view the required statistics course as an intimidating obstacle that has been placed in the middle of an otherwise interesting curriculum. They want to learn about human behavior—not about math and science. As a result, the statistics course is seen as irrelevant to their education and career goals. However, as long as the behavioral sciences are founded in science, knowledge of statistics will be necessary. Statistical procedures provide researchers with objective and systematic methods for describing and interpreting their research results. Scientific research is the system that we use to gather information, and statistics are the tools that we use to distill the information into sensible and justified conclusions. The goal of this book is not only to teach the methods of statistics, but also to convey the basic principles of objectivity and logic that are essential for science and valuable for decision making in everyday life. Those of you who are familiar with previous editions of Statistics for the Behavioral Sciences will notice that some changes have been made. These changes are summarized in the section entitled “To the Instructor.” In revising this text, our students have been foremost in our minds. Over the years, they have provided honest and useful feedback. Their hard work and perseverance has made our writing and teaching most rewarding. We sincerely thank them. Students who are using this edition should please read the section of the preface entitled “To the Student.” The book chapters are organized in the sequence that we use for our own statistics courses. We begin with descriptive statistics, and then examine a variety of statistical procedures focused on sample means and variance before moving on to correlational methods and nonparametric statistics. Information about modifying this sequence is presented in the To The Instructor section for individuals who prefer a different organization. Each chapter contains numerous examples, many based on actual research studies, learning checks, a summary and list of key terms, and a set of 20–30 problems. Ancillaries Ancillaries for this edition include the following. ■■ MindTap® Psychology: MindTap® Psychology for Gravetter/Wallnau’s Statistics for The Behavioral Sciences, 10th Edition is the digital learning solution that helps instructors engage and transform today’s students into critical thinkers. Through paths of dynamic assignments and applications that you can personalize, real-time course analytics, and an accessible reader, MindTap helps you turn cookie cutter into cutting edge, apathy into engagement, and memorizers into higher-level thinkers. As an instructor using MindTap you have at your fingertips the right content and unique set of tools curated specifically for your course, such as video tutorials that walk students through various concepts and interactive problem tutorials that provide students opportunities to practice what they have learned, all in an interface designed to improve workflow and save time when planning lessons and course structure. The control to build and personalize your course is all yours, focusing on the most relevant xiii xiv PREFACE ■■ ■■ ■■ material while also lowering costs for your students. Stay connected and informed in your course through real time student tracking that provides the opportunity to adjust the course as needed based on analytics of interactivity in the course. Online Instructor’s Manual: The manual includes learning objectives, key terms, a detailed chapter outline, a chapter summary, lesson plans, discussion topics, student activities, “What If” scenarios, media tools, a sample syllabus and an expanded test bank. The learning objectives are correlated with the discussion topics, student activities, and media tools. Online PowerPoints: Helping you make your lectures more engaging while effectively reaching your visually oriented students, these handy Microsoft PowerPoint® slides outline the chapters of the main text in a classroom-ready presentation. The PowerPoint® slides are updated to reflect the content and organization of the new edition of the text. Cengage Learning Testing, powered by Cognero®: Cengage Learning Testing, Powered by Cognero®, is a flexible online system that allows you to author, edit, and manage test bank content. You can create multiple test versions in an instant and deliver tests from your LMS in your classroom. Acknowledgments It takes a lot of good, hard-working people to produce a book. Our friends at Cengage have made enormous contributions to this textbook. We thank: Jon-David Hague, Product Director; Timothy Matray, Product Team Director; Jasmin Tokatlian, Content Development Manager; Kimiya Hojjat, Product Assistant; and Vernon Boes, Art Director. Special thanks go to Stefanie Chase, our Content Developer and to Lynn Lustberg who led us through production at MPS. Reviewers play a very important role in the development of a manuscript. Accordingly, we offer our appreciation to the following colleagues for their assistance: Patricia Case, University of Toledo; Kevin David, Northeastern State University; Adia Garrett, University of Maryland, Baltimore County; Carrie E. Hall, Miami University; Deletha Hardin, University of Tampa; Angela Heads, Prairie View A&M University; Roberto Heredia, Texas A&M International University; Alisha Janowski, University of Central Florida; Matthew Mulvaney, The College at Brockport (SUNY); Nicholas Von Glahn, California State Polytechnic University, Pomona; and Ronald Yockey, Fresno State University. To the Instructor Those of you familiar with the previous edition of Statistics for the Behavioral Sciences will notice a number of changes in the 10th edition. Throughout this book, research examples have been updated, real world examples have been added, and the end-of-chapter problems have been extensively revised. Major revisions for this edition include the following: 1. Each section of every chapter begins with a list of Learning Objectives for that specific section. 2. Each section ends with a Learning Check consisting of multiple-choice questions with at least one question for each Learning Objective. PREFACE xv 3. The former Chapter 19, Choosing the Right Statistics, has been eliminated and an abridged version is now an Appendix replacing the Statistics Organizer, which appeared in earlier editions. Other examples of specific and noteworthy revisions include the following. Chapter 1 The section on data structures and research methods parallels the new Appendix, Choosing the Right Statistics. Chapter 2 The chapter opens with a new Preview to introduce the concept and purpose of frequency distributions. Chapter 3 Minor editing clarifies and simplifies the discussion the median. Chapter 4 The chapter opens with a new Preview to introduce the topic of Central Tendency. The sections on standard deviation and variance have been edited to increase emphasis on concepts rather than calculations. The section discussion relationships between z, X, μ, and σ has been expanded and includes a new demonstration example. Chapter 5 Chapter 6 The chapter opens with a new Preview to introduce the topic of Probability. The section, Looking Ahead to Inferential Statistics, has been substantially shortened and simplified. Chapter 7 The former Box explaining difference between standard deviation and standard error was deleted and the content incorporated into Section 7.4 with editing to emphasize that the standard error is the primary new element introduced in the chapter. The final section, Looking Ahead to Inferential Statistics, was simplified and shortened to be consistent with the changes in Chapter 6. Chapter 8 A redundant example was deleted which shortened and streamlined the remaining material so that most of the chapter is focused on the same research example. Chapter 9 The chapter opens with a new Preview to introduce the t statistic and explain why a new test statistic is needed. The section introducing Confidence Intervals was edited to clarify the origin of the confidence interval equation and to emphasize that the interval is constructed at the sample mean. Chapter 10 The chapter opens with a new Preview introducing the independent-measures t statistic. The section presenting the estimated standard error of (M1 – M2) has been simplified and shortened. Chapter 11 The chapter opens with a new Preview introducing the repeated-measures t statistic. The section discussing hypothesis testing has been separated from the section on effect size and confidence intervals to be consistent with the other two chapters on t tests. The section comparing independent- and repeated-measures designs has been expanded. Chapter 12 The chapter opens with a new Preview introducing ANOVA and explaining why a new hypothesis testing procedure is necessary. Sections in the chapter have been reorganized to allow flow directly from hypothesis tests and effect size to post tests. xvi PREFACE Chapter 13 Substantially expanded the section discussing factors that influence the outcome of a repeated-measures hypothesis test and associated measures of effect size. Chapter 14 The chapter opens with a new Preview presenting a two-factor research example and introducing the associated ANOVA. Sections have been reorganized so that simple main effects and the idea of using a second factor to reduce variance from individual differences are now presented as extra material related to the two-factor ANOVA. Chapter 15 The chapter opens with a new Preview presenting a correlational research study and the concept of a correlation. A new section introduces the t statistic for evaluating the significance of a correlation and the section on partial correlations has been simplified and shortened. Chapter 16 The chapter opens with a new Preview introducing the concept of regression and its purpose. A new section demonstrates the equivalence of testing the significance of a correlation and testing the significance of a regression equation with one predictor variable. The section on residuals for the multiple-regression equation has been edited to simplify and shorten. Chapter 17 A new chapter Preview presents an experimental study with data consisting of frequencies, which are not compatible with computing means and variances. Chi-square tests are introduced as a solution to this problem. A new section introduces Cohen’s w as a means of measuring effect size for both chi-square tests. Chapter 18 Substantial editing clarifies the section explaining how the real limits for each score can influence the conclusion from a binomial test. The former Chapter 19 covering the task of matching statistical methods to specific types of data has been substantially shortened and converted into an Appendix. ■■Matching the Text to Your Syllabus The book chapters are organized in the sequence that we use for our own statistics courses. However, different instructors may prefer different organizations and probably will choose to omit or deemphasize specific topics. We have tried to make separate chapters, and even sections of chapters, completely self-contained, so they can be deleted or reorganized to fit the syllabus for nearly any instructor. Some common examples are as follows. ■■ ■■ ■■ It is common for instructors to choose between emphasizing analysis of variance (Chapters 12, 13, and 14) or emphasizing correlation/regression (Chapters 15 and 16). It is rare for a one-semester course to complete coverage of both topics. Although we choose to complete all the hypothesis tests for means and mean differences before introducing correlation (Chapter 15), many instructors prefer to place correlation much earlier in the sequence of course topics. To accommodate this, Sections 15.1, 15.2, and 15.3 present the calculation and interpretation of the Pearson correlation and can be introduced immediately following Chapter 4 (variability). Other sections of Chapter 15 refer to hypothesis testing and should be delayed until the process of hypothesis testing (Chapter 8) has been introduced. It is also possible for instructors to present the chi-square tests (Chapter 17) much earlier in the sequence of course topics. Chapter 17, which presents hypothesis tests for proportions, can be presented immediately after Chapter 8, which introduces the process of hypothesis testing. If this is done, we also recommend that the Pearson correlation (Sections 15.1, 15.2, and 15.3) be presented early to provide a foundation for the chi-square test for independence. PREFACE xvii To the Student A primary goal of this book is to make the task of learning statistics as easy and painless as possible. Among other things, you will notice that the book provides you with a number of opportunities to practice the techniques you will be learning in the form of Learning Checks, Examples, Demonstrations, and end-of-chapter problems. We encourage you to take advantage of these opportunities. Read the text rather than just memorizing the formulas. We have taken care to present each statistical procedure in a conceptual context that explains why the procedure was developed and when it should be used. If you read this material and gain an understanding of the basic concepts underlying a statistical formula, you will find that learning the formula and how to use it will be much easier. In the “Study Hints,” that follow, we provide advice that we give our own students. Ask your instructor for advice as well; we are sure that other instructors will have ideas of their own. Over the years, the students in our classes and other students using our book have given us valuable feedback. If you have any suggestions or comments about this book, you can write to either Professor Emeritus Frederick Gravetter or Professor Emeritus Larry Wallnau at the Department of Psychology, SUNY College at Brockport, 350 New Campus Drive, Brockport, New York 14420. You can also contact Professor Emeritus Gravetter directly at fgravett@brockport.edu. ■■Study Hints You may find some of these tips helpful, as our own students have reported. ■■ ■■ ■■ ■■ ■■ The key to success in a statistics course is to keep up with the material. Each new topic builds on previous topics. If you have learned the previous material, then the new topic is just one small step forward. Without the proper background, however, the new topic can be a complete mystery. If you find that you are falling behind, get help immediately. You will learn (and remember) much more if you study for short periods several times per week rather than try to condense all of your studying into one long session. For example, it is far more effective to study half an hour every night than to have a single 3½-hour study session once a week. We cannot even work on writing this book without frequent rest breaks. Do some work before class. Keep a little ahead of the instructor by reading the appropriate sections before they are presented in class. Although you may not fully understand what you read, you will have a general idea of the topic, which will make the lecture easier to follow. Also, you can identify material that is particularly confusing and then be sure the topic is clarified in class. Pay attention and think during class. Although this advice seems obvious, often it is not practiced. Many students spend so much time trying to write down every example presented or every word spoken by the instructor that they do not actually understand and process what is being said. Check with your instructor—there may not be a need to copy every example presented in class, especially if there are many examples like it in the text. Sometimes, we tell our students to put their pens and pencils down for a moment and just listen. Test yourself regularly. Do not wait until the end of the chapter or the end of the week to check your knowledge. After each lecture, work some of the end-of-chapter problems and do the Learning Checks. Review the Demonstration Problems, and be sure you can define the Key Terms. If you are having trouble, get your questions answered immediately—reread the section, go to your instructor, or ask questions in class. By doing so, you will be able to move ahead to new material. xviii PREFACE ■■ ■■ Do not kid yourself! Avoid denial. Many students observe their instructor solve problems in class and think to themselves, “This looks easy, I understand it.” Do you really understand it? Can you really do the problem on your own without having to leaf through the pages of a chapter? Although there is nothing wrong with using examples in the text as models for solving problems, you should try working a problem with your book closed to test your level of mastery. We realize that many students are embarrassed to ask for help. It is our biggest challenge as instructors. You must find a way to overcome this aversion. Perhaps contacting the instructor directly would be a good starting point, if asking questions in class is too anxiety-provoking. You could be pleasantly surprised to find that your instructor does not yell, scold, or bite! Also, your instructor might know of another student who can offer assistance. Peer tutoring can be very helpful. Frederick J Gravetter Larry B. Wallnau A B o U t tH E AU tH o R S Frederick Gravetter Frederick J Gravetter is Professor Emeritus of Psychology at the State University of New York College at Brockport. While teaching at Brockport, Dr. Gravetter specialized in statistics, experimental design, and cognitive psychology. He received his bachelor’s degree in mathematics from M.I.T. and his Ph.D in psychology from Duke University. In addition to publishing this textbook and several research articles, Dr. Gravetter co-authored Research Methods for the Behavioral Science and Essentials of Statistics for the Behavioral Sciences. Larry B. Wallnau Larry B. WaLLnau is Professor Emeritus of Psychology at the State University of New York College at Brockport. While teaching at Brockport, he published numerous research articles in biopsychology. With Dr. Gravetter, he co-authored Essentials of Statistics for the Behavioral Sciences. Dr. Wallnau also has provided editorial consulting for numerous publishers and journals. He has taken up running and has competed in 5K races in New York and Connecticut. He takes great pleasure in adopting neglected and rescued dogs. xix CH A P T ER Introduction to Statistics 1 © Deborah Batt PREVIEW 1.1 Statistics, Science, and Observations 1.2 Data Structures, Research Methods, and Statistics 1.3 Variables and Measurement 1.4 Statistical Notation Summary Focus on Problem Solving Demonstration 1.1 Problems 1 PREVIEW Before we begin our discussion of statistics, we ask you to read the following paragraph taken from the philosophy of Wrong Shui (Candappa, 2000). The Journey to Enlightenment In Wrong Shui, life is seen as a cosmic journey, a struggle to overcome unseen and unexpected obstacles at the end of which the traveler will find illumination and enlightenment. Replicate this quest in your home by moving light switches away from doors and over to the far side of each room.* Why did we begin a statistics book with a bit of twisted philosophy? In part, we simply wanted to lighten the mood with a bit of humor—starting a statistics course is typically not viewed as one of life’s joyous moments. In addition, the paragraph is an excellent counterexample for the purpose of this book. Specifically, our goal is to do everything possible to prevent you from stumbling around in the dark by providing lots of help and illumination as you journey through the world of statistics. To accomplish this, we begin each section of the book with clearly stated learning objectives and end each section with a brief quiz to test your mastery of the new material. We also introduce each new statistical procedure by explaining the purpose it is intended to serve. If you understand why a new procedure is needed, you will find it much easier to learn. 1.1 The objectives for this first chapter are to provide an introduction to the topic of statistics and to give you some background for the rest of the book. We discuss the role of statistics within the general field of scientific inquiry, and we introduce some of the vocabulary and notation that are necessary for the statistical methods that follow. As you read through the following chapters, keep in mind that the general topic of statistics follows a well-organized, logically developed progression that leads from basic concepts and definitions to increasingly sophisticated techniques. Thus, each new topic serves as a foundation for the material that follows. The content of the first nine chapters, for example, provides an essential background and context for the statistical methods presented in Chapter 10. If you turn directly to Chapter 10 without reading the first nine chapters, you will find the material confusing and incomprehensible. However, if you learn and use the background material, you will have a good frame of reference for understanding and incorporating new concepts as they are presented. *Candappa, R. (2000). The little book of wrong shui. Kansas City: Andrews McMeel Publishing. Reprinted by permission. Statistics, Science, and Observations LEARNING OBJECTIVEs 1. Define the terms population, sample, parameter, and statistic, and describe the relationships between them. 2. Define descriptive and inferential statistics and describe how these two general categories of statistics are used in a typical research study. 3. Describe the concept of sampling error and explain how this concept creates the fundamental problem that inferential statistics must address. ■■Definitions of Statistics By one definition, statistics consist of facts and figures such as the average annual snowfall in Denver or Derrick Jeter’s lifetime batting average. These statistics are usually informative and time-saving because they condense large quantities of information into a few simple figures. Later in this chapter we return to the notion of calculating statistics (facts and figures) but, for now, we concentrate on a much broader definition of statistics. Specifically, we use the term statistics to refer to a general field of mathematics. In this case, we are using the term statistics as a shortened version of statistical procedures. For example, you are probably using this book for a statistics course in which you will learn about the statistical techniques that are used to summarize and evaluate research results in the behavioral sciences. 2 SEctIon 1.1 | Statistics, Science, and Observations 3 Research in the behavioral sciences (and other fields) involves gathering information. To determine, for example, whether college students learn better by reading material on printed pages or on a computer screen, you would need to gather information about students’ study habits and their academic performance. When researchers finish the task of gathering information, they typically find themselves with pages and pages of measurements such as preferences, personality scores, opinions, and so on. In this book, we present the statistics that researchers use to analyze and interpret the information that they gather. Specifically, statistics serve two general purposes: 1. Statistics are used to organize and summarize the information so that the researcher can see what happened in the research study and can communicate the results to others. 2. Statistics help the researcher to answer the questions that initiated the research by determining exactly what general conclusions are justified based on the specific results that were obtained. DEFInItIon The term statistics refers to a set of mathematical procedures for organizing, summarizing, and interpreting information. Statistical procedures help ensure that the information or observations are presented and interpreted in an accurate and informative way. In somewhat grandiose terms, statistics help researchers bring order out of chaos. In addition, statistics provide researchers with a set of standardized techniques that are recognized and understood throughout the scientific community. Thus, the statistical methods used by one researcher will be familiar to other researchers, who can accurately interpret the statistical analyses with a full understanding of how the analysis was done and what the results signify. ■■Populations and Samples Research in the behavioral sciences typically begins with a general question about a specific group (or groups) of individuals. For example, a researcher may want to know what factors are associated with academic dishonesty among college students. Or a researcher may want to examine the amount of time spent in the bathroom for men compared to women. In the first example, the researcher is interested in the group of college students. In the second example, the researcher wants to compare the group of men with the group of women. In statistical terminology, the entire group that a researcher wishes to study is called a population. DEFInItIon A population is the set of all the individuals of interest in a particular study. As you can well imagine, a population can be quite large—for example, the entire set of women on the planet Earth. A researcher might be more specific, limiting the population for study to women who are registered voters in the United States. Perhaps the investigator would like to study the population consisting of women who are heads of state. Populations can obviously vary in size from extremely large to very small, depending on how the investigator defines the population. The population being studied should always be identified by the researcher. In addition, the population need not consist of people—it could be a population of rats, corporations, parts produced in a factory, or anything else an investigator wants to study. In practice, populations are typically very large, such as the population of college sophomores in the United States or the population of small businesses. Because populations tend to be very large, it usually is impossible for a researcher to examine every individual in the population of interest. Therefore, researchers typically select 4 chaPtER 1 | Introduction to Statistics a smaller, more manageable group from the population and limit their studies to the individuals in the selected group. In statistical terms, a set of individuals selected from a population is called a sample. A sample is intended to be representative of its population, and a sample should always be identified in terms of the population from which it was selected. A sample is a set of individuals selected from a population, usually intended to represent the population in a research study. DEFInItIon Just as we saw with populations, samples can vary in size. For example, one study might examine a sample of only 10 students in a graduate program and another study might use a sample of more than 10,000 people who take a specific cholesterol medication. So far we have talked about a sample being selected from a population. However, this is actually only half of the full relationship between a sample and its population. Specifically, when a researcher finishes examining the sample, the goal is to generalize the results back to the entire population. Remember that the research started with a general question about the population. To answer the question, a researcher studies a sample and then generalizes the results from the sample to the population. The full relationship between a sample and a population is shown in Figure 1.1. ■■Variables and Data Typically, researchers are interested in specific characteristics of the individuals in the population (or in the sample), or they are interested in outside factors that may influence the individuals. For example, a researcher may be interested in the influence of the weather on people’s moods. As the weather changes, do people’s moods also change? Something that can change or have different values is called a variable. DEFInItIon A variable is a characteristic or condition that changes or has different values for different individuals. THE POPULATION All of the individuals of interest The results from the sample are generalized to the population F I G U R E 1.1 The relationship between a population and a sample. The sample is selected from the population THE SAMPLE The individuals selected to participate in the research study SEctIon 1.1 | Statistics, Science, and Observations 5 Once again, variables can be characteristics that differ from one individual to another, such as height, weight, gender, or personality. Also, variables can be environmental conditions that change such as temperature, time of day, or the size of the room in which the research is being conducted. To demonstrate changes in variables, it is necessary to make measurements of the variables being examined. The measurement obtained for each individual is called a datum, or more commonly, a score or raw score. The complete set of scores is called the data set or simply the data. DEFInItIon Data (plural) are measurements or observations. A data set is a collection of measurements or observations. A datum (singular) is a single measurement or observation and is commonly called a score or raw score. Before we move on, we should make one more point about samples, populations, and data. Earlier, we defined populations and samples in terms of individuals. For example, we discussed a population of graduate students and a sample of cholesterol patients. Be forewarned, however, that we will also refer to populations or samples of scores. Because research typically involves measuring each individual to obtain a score, every sample (or population) of individuals produces a corresponding sample (or population) of scores. ■■Parameters and Statistics When describing data it is necessary to distinguish whether the data come from a population or a sample. A characteristic that describes a population—for example, the average score for the population—is called a parameter. A characteristic that describes a sample is called a statistic. Thus, the average score for a sample is an example of a statistic. Typically, the research process begins with a question about a population parameter. However, the actual data come from a sample and are used to compute sample statistics. DEFInItIon A parameter is a value, usually a numerical value, that describes a population. A parameter is usually derived from measurements of the individuals in the population. A statistic is a value, usually a numerical value, that describes a sample. A statistic is usually derived from measurements of the individuals in the sample. Every population parameter has a corresponding sample statistic, and most research studies involve using statistics from samples as the basis for answering questions about population parameters. As a result, much of this book is concerned with the relationship between sample statistics and the corresponding population parameters. In Chapter 7, for example, we examine the relationship between the mean obtained for a sample and the mean for the population from which the sample was obtained. ■■Descriptive and Inferential Statistical Methods Although researchers have developed a variety of different statistical procedures to organize and interpret data, these different procedures can be classified into two general categories. The first category, descriptive statistics, consists of statistical procedures that are used to simplify and summarize data. DEFInItIon Descriptive statistics are statistical procedures used to summarize, organize, and simplify data. 6 chaPtER 1 | Introduction to Statistics Descriptive statistics are techniques that take raw scores and organize or summarize them in a form that is more manageable. Often the scores are organized in a table or a graph so that it is possible to see the entire set of scores. Another common technique is to summarize a set of scores by computing an average. Note that even if the data set has hundreds of scores, the average provides a single descriptive value for the entire set. The second general category of statistical techniques is called inferential statistics. Inferential statistics are methods that use sample data to make general statements about a population. DEFInItIon Inferential statistics consist of techniques that allow us to study samples and then make generalizations about the populations from which they were selected. Because populations are typically very large, it usually is not possible to measure everyone in the population. Therefore, a sample is selected to represent the population. By analyzing the results from the sample, we hope to make general statements about the population. Typically, researchers use sample statistics as the basis for drawing conclusions about population parameters. One problem with using samples, however, is that a sample provides only limited information about the population. Although samples are generally representative of their populations, a sample is not expected to give a perfectly accurate picture of the whole population. There usually is some discrepancy between a sample statistic and the corresponding population parameter. This discrepancy is called sampling error, and it creates the fundamental problem inferential statistics must always address. DEFInItIon Sampling error is the naturally occurring discrepancy, or error, that exists between a sample statistic and the corresponding population parameter. The concept of sampling error is illustrated in Figure 1.2. The figure shows a population of 1,000 college students and 2 samples, each with 5 students who were selected from the population. Notice that each sample contains different individuals who have different characteristics. Because the characteristics of each sample depend on the specific people in the sample, statistics will vary from one sample to another. For example, the five students in sample 1 have an average age of 19.8 years and the students in sample 2 have an average age of 20.4 years. It is also very unlikely that the statistics obtained for a sample will be identical to the parameters for the entire population. In Figure 1.2, for example, neither sample has statistics that are exactly the same as the population parameters. You should also realize that Figure 1.2 shows only two of the hundreds of possible samples. Each sample would contain different individuals and would produce different statistics. This is the basic concept of sampling error: sample statistics vary from one sample to another and typically are different from the corresponding population parameters. One common example of sampling error is the error associated with a sample proportion. For example, in newspaper articles reporting results from political polls, you frequently find statements such as this: Candidate Brown leads the poll with 51% of the vote. Candidate Jones has 42% approval, and the remaining 7% are undecided. This poll was taken from a sample of registered voters and has a margin of error of plus-or-minus 4 percentage points. The “margin of error” is the sampling error. In this case, the percentages that are reported were obtained from a sample and are being generalized to the whole population. As always, you do not expect the statistics from a sample to be perfect. There always will be some “margin of error” when sample statistics are used to represent population parameters. SEctIon 1.1 | Statistics, Science, and Observations 7 F I G U R E 1. 2 A demonstration of sampling error. Two samples are selected from the same population. Notice that the sample statistics are different from one sample to another and all the sample statistics are different from the corresponding population parameters. The natural differences that exist, by chance, between a sample statistic and population parameter are called sampling error. Population of 1000 college students Population Parameters Average Age 5 21.3 years Average IQ 5 112.5 65% Female, 35% Male Sample #1 Sample #2 Eric Jessica Laura Karen Brian Tom Kristen Sara Andrew John Sample Statistics Average Age 5 19.8 Average IQ 5 104.6 60% Female, 40% Male Sample Statistics Average Age 5 20.4 Average IQ 5 114.2 40% Female, 60% Male As a further demonstration of sampling error, imagine that your statistics class is separated into two groups by drawing a line from front to back through the middle of the room. Now imagine that you compute the average age (or height, or IQ) for each group. Will the two groups have exactly the same average? Almost certainly they will not. No matter what you chose to measure, you will probably find some difference between the two groups. However, the difference you obtain does not necessarily mean that there is a systematic difference between the two groups. For example, if the average age for students on the right-hand side of the room is higher than the average for students on the left, it is unlikely that some mysterious force has caused the older people to gravitate to the right side of the room. Instead, the difference is probably the result of random factors such as chance. The unpredictable, unsystematic differences that exist from one sample to another are an example of sampling error. ■■Statistics in the Context of Research The following example shows the general stages of a research study and demonstrates how descriptive statistics and inferential statistics are used to organize and interpret the data. At the end of the example, note how sampling error can affect the interpretation of experimental results, and consider why inferential statistical methods are needed to deal with this problem. 8 chaPtER 1 | Introduction to Statistics ExamplE 1.1 Figure 1.3 shows an overview of a general research situation and demonstrates the roles that descriptive and inferential statistics play. The purpose of the research study is to address a question that we posed earlier: Do college students learn better by studying text on printed pages or on a computer screen? Two samples are selected from the population of college students. The students in sample A are given printed pages of text to study for 30 minutes and the students in sample B study the same text on a computer screen. Next, all of the students are given a multiple-choice test to evaluate their knowledge of the material. At this point, the researcher has two sets of data: the scores for sample A and the scores for sample B (see the figure). Now is the time to begin using statistics. First, descriptive statistics are used to simplify the pages of data. For example, the researcher could draw a graph showing the scores for each sample or compute the average score for each sample. Note that descriptive methods provide a simplified, organized Step 1 Experiment: Compare two studying methods Data Test scores for the students in each sample Step 2 Descriptive statistics: Organize and simplify Population of College Students Sample A Read from printed pages 25 27 30 19 29 26 21 28 23 26 20 25 28 27 24 26 22 30 Average Score = 26 Step 3 Inferential statistics: Interpret results F i g u r E 1. 3 The role of statistics in experimental research. Sample B Read from computer screen 20 20 23 25 22 18 22 17 28 19 24 25 30 27 23 21 22 19 Average Score = 22 The sample data show a 4-point difference between the two methods of studying. However, there are two ways to interpret the results. 1. There actually is no difference between the two studying methods, and the sample difference is due to chance (sampling error). 2. There really is a difference between the two methods, and the sample data accurately reflect this difference. The goal of inferential statistics is to help researchers decide between the two interpretations. SEctIon 1.1 | Statistics, Science, and Observations 9 description of the scores. In this example, the students who studied printed pages had an average score of 26 on the test, and the students who studied text on the computer averaged 22. Once the researcher has described the results, the next step is to interpret the outcome. This is the role of inferential statistics. In this example, the researcher has found a difference of 4 points between the two samples (sample A averaged 26 and sample B averaged 22). The problem for inferential statistics is to differentiate between the following two interpretations: 1. There is no real difference between the printed page and a computer screen, and the 4-point difference between the samples is just an example of sampling error (like the samples in Figure 1.2). 2. There really is a difference between the printed page and a computer screen, and the 4-point difference between the samples was caused by the different methods of studying. In simple English, does the 4-point difference between samples provide convincing evidence of a difference between the two studying methods, or is the 4-point difference just chance? The purpose of inferential statistics is to answer this question. ■ lE arn in g Ch ECk 1. A researcher is interested in the sleeping habits of American college students. A group of 50 students is interviewed and the researcher finds that these students sleep an average of 6.7 hours per day. For this study, the average of 6.7 hours is an example of a(n) . a. parameter b. statistic c. population d. sample 2. A researcher is curious about the average IQ of registered voters in the state of Florida. The entire group of registered voters in the state is an example of a . a. sample b. statistic c. population d. parameter 3. Statistical techniques that summarize, organize, and simplify data are classified as . a. population statistics b. sample statistics c. descriptive statistics d. inferential statistics 4. In general, statistical techniques are used to summarize the data from a research study and statistical techniques are used to determine what conclusions are justified by the results. a. inferential, descriptive b. descriptive, inferential c. sample, population d. population, sample 10 chaPtER 1 | Introduction to Statistics 5. IQ tests are standardized so that the average score is 100 for the entire group of people who take the test each year. However, if you selected a group of 20 people who took the test and computed their average IQ score you probably would not get 100. What statistical concept explains the difference between your mean and the mean for the entire group? a. statistical error b. inferential error c. descriptive error d. sampling error an s wE r s 1. B, 2. C, 3. C, 4. B, 5. D 1.2 Data Structures, Research Methods, and Statistics LEARNING OBJECTIVEs 4. Differentiate correlational, experimental, and nonexperimental research and describe the data structures associated with each. 5. Define independent, dependent, and quasi-independent variables and recognize examples of each. ■■Individual Variables: Descriptive Research Some research studies are conducted simply to describe individual variables as they exist naturally. For example, a college official may conduct a survey to describe the eating, sleeping, and study habits of a group of college students. When the results consist of numerical scores, such as the number of hours spent studying each day, they are typically described by the statistical techniques that are presented in Chapters 3 and 4. Non-numerical scores are typically described by computing the proportion or percentage in each category. For example, a recent newspaper article reported that 34.9% of Americans are obese, which is roughly 35 pounds over a healthy weight. ■■Relationships Between Variables Most research, however, is intended to examine relationships between two or more variables. For example, is there a relationship between the amount of violence in the video games played by children and the amount of aggressive behavior they display? Is there a relationship between the quality of breakfast and academic performance for elementary school children? Is there a relationship between the number of hours of sleep and grade point average for college students? To establish the existence of a relationship, researchers must make observations—that is, measurements of the two variables. The resulting measurements can be classified into two distinct data structures that also help to classify different research methods and different statistical techniques. In the following section we identify and discuss these two data structures. I. One Group with Two Variables Measured for Each Individual: The Correlational Method One method for examining the relationship between variables is to observe the two variables as they exist naturally for a set of individuals. That is, simply SEctIon 1.2 | Data Structures, Research Methods, and Statistics 11 measure the two variables for each individual. For example, research has demonstrated a relationship between sleep habits, especially wake-up time, and academic performance for college students (Trockel, Barnes, and Egget, 2000). The researchers used a survey to measure wake-up time and school records to measure academic performance for each student. Figure 1.4 shows an example of the kind of data obtained in the study. The researchers then look for consistent patterns in the data to provide evidence for a relationship between variables. For example, as wake-up time changes from one student to another, is there also a tendency for academic performance to change? Consistent patterns in the data are often easier to see if the scores are presented in a graph. Figure 1.4 also shows the scores for the eight students in a graph called a scatter plot. In the scatter plot, each individual is represented by a point so that the horizontal position corresponds to the student’s wake-up time and the vertical position corresponds to the student’s academic performance score. The scatter plot shows a clear relationship between wake-up time and academic performance: as wake-up time increases, academic performance decreases. A research study that simply measures two different variables for each individual and produces the kind of data shown in Figure 1.4 is an example of the correlational method, or the correlational research strategy. In the correlational method, two different variables are observed to determine whether there is a relationship between them. DEFInItIon ■■Statistics for the Correlational Method When the data from a correlational study consist of numerical scores, the relationship between the two variables is usually measured and described using a statistic called a correlation. Correlations and the correlational method are discussed in detail in Chapters 15 and 16. Occasionally, the measurement process used for a correlational study simply classifies individuals into categories that do not correspond to numerical values. For example, a researcher could classify a group of college students by gender (male Student Wake-up Time Academic Performance A B C D E F G H 11 9 9 12 7 10 10 8 2.4 3.6 3.2 2.2 3.8 2.2 3.0 3.0 (b) Academic performance (a) 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 7 F i g u r E 1. 4 8 9 10 11 12 Wake-up time One of two data structures for evaluating the relationship between variables. Note that there are two separate measurements for each individual (wake-up time and academic performance). The same scores are shown in a table (a) and in a graph (b). 12 chaPtER 1 | Introduction to Statistics or female) and by cell-phone preference (talk or text). Note that the researcher has two scores for each individual but neither of the scores is a numerical value. This type of data is typically summarized in a table showing how many individuals are classified into each of the possible categories. Table 1.1 shows an example of this kind of summary table. The table shows for example, that 30 of the males in the sample preferred texting to talking. This type of data can be coded with numbers (for example, male = 0 and female = 1) so that it is possible to compute a correlation. However, the relationship between variables for non-numerical data, such as the data in Table 1.1, is usually evaluated using a statistical technique known as a chi-square test. Chi-square tests are presented in Chapter 17. Ta b lE 1.1 Correlational data consisting of non-numerical scores. Note that there are two measurements for each individual: gender and cell phone preference. The numbers indicate how many people are in each category. For example, out of the 50 males, 30 prefer text over talk. Cell Phone Preference Text Talk Males 30 20 50 Females 25 25 50 ■■Limitations of the Correlational Method The results from a correlational study can demonstrate the existence of a relationship between two variables, but they do not provide an explanation for the relationship. In particular, a correlational study cannot demonstrate a cause-and-effect relationship. For example, the data in Figure 1.4 show a systematic relationship between wake-up time and academic performance for a group of college students; those who sleep late tend to have lower performance scores than those who wake early. However, there are many possible explanations for the relationship and we do not know exactly what factor (or factors) is responsible for late sleepers having lower grades. In particular, we cannot conclude that waking students up earlier would cause their academic performance to improve, or that studying more would cause students to wake up earlier. To demonstrate a cause-and-effect relationship between two variables, researchers must use the experimental method, which is discussed next. II. Comparing Two (or More) Groups of Scores: Experimental and Nonexperimental Methods The second method for examining the relationship between two variables involves the comparison of two or more groups of scores. In this situation, the relationship between variables is examined by using one of the variables to define the groups, and then measuring the second variable to obtain scores for each group. For example, Polman, de Castro, and van Aken (2008) randomly divided a sample of 10-year-old boys into two groups. One group then played a violent video game and the second played a nonviolent game. After the game-playing session, the children went to a free play period and were monitored for aggressive behaviors (hitting, kicking, pushing, frightening, name calling, fighting, quarreling, or teasing another child). An example of the resulting data is shown in Figure 1.5. The researchers then compare the scores for the violent-video group with the scores for the nonviolent-video group. A systematic difference between the two groups provides evidence for a relationship between playing violent video games and aggressive behavior for 10-year-old boys. SEctIon 1.2 | Data Structures, Research Methods, and Statistics F i g u r E 1. 5 Evaluating the relationship between variables by comparing groups of scores. Note that the values of one variable are used to define the groups and the second variable is measured to obtain scores within each group. One variable (type of video game) is used to define groups A second variable (aggressive behavior) is measured to obtain scores within each group Violent Nonviolent 7 8 10 7 9 8 6 10 9 6 8 4 8 3 6 5 3 4 4 5 13 Compare groups of scores ■■Statistics for Comparing Two (or More) Groups of Scores Most of the statistical procedures presented in this book are designed for research studies that compare groups of scores like the study in Figure 1.5. Specifically, we examine descriptive statistics that summarize and describe the scores in each group and we use inferential statistics to determine whether the differences between the groups can be generalized to the entire population. When the measurement procedure produces numerical scores, the statistical evaluation typically involves computing the average score for each group and then comparing the averages. The process of computing averages is presented in Chapter 3, and a variety of statistical techniques for comparing averages are presented in Chapters 8–14. If the measurement process simply classifies individuals into non-numerical categories, the statistical evaluation usually consists of computing proportions for each group and then comparing proportions. Previously, in Table 1.1, we presented an example of non-numerical data examining the relationship between gender and cell-phone preference. The same data can be used to compare the proportions for males with the proportions for females. For example, using text is preferred by 60% of the males compared to 50% of the females. As before, these data are evaluated using a chi-square test, which is presented in Chapter 17. ■■Experimental and Nonexperimental Methods There are two distinct research methods that both produce groups of scores to be compared: the experimental and the nonexperimental strategies. These two research methods use exactly the same statistics and they both demonstrate a relationship between two variables. The distinction between the two research strategies is how the relationship is interpreted. The results from an experiment allow a cause-and-effect explanation. For example, we can conclude that changes in one variable are responsible for causing differences in a second variable. A nonexperimental study does not permit a cause-and effect explanation. We can say that changes in one variable are accompanied by changes in a second variable, but we cannot say why. Each of the two research methods is discussed in the following sections. ■■The Experimental Method One specific research method that involves comparing groups of scores is known as the experimental method or the experimental research strategy. The goal of an experimental study is to demonstrate a cause-and-effect relationship between two variables. Specifically, 14 chaPtER 1 | Introduction to Statistics an experiment attempts to show that changing the value of one variable causes changes to occur in the second variable. To accomplish this goal, the experimental method has two characteristics that differentiate experiments from other types of research studies: 1. Manipulation The researcher manipulates one variable by changing its value from one level to another. In the Polman et al. (2008) experiment examining the effect of violence in video games (Figure 1.5), the researchers manipulate the amount of violence by giving one group of boys a violent game to play and giving the other group a nonviolent game. A second variable is observed (measured) to determine whether the manipulation causes changes to occur. 2. Control The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined. In more complex experiments, a researcher may systematically manipulate more than one variable and may observe more than one variable. Here we are considering the simplest case, in which only one variable is manipulated and only one variable is observed. To demonstrate these two characteristics, consider the Polman et al. (2008) study examining the effect of violence in video games (see Figure 1.5). To be able to say that the difference in aggressive behavior is caused by the amount of violence in the game, the researcher must rule out any other possible explanation for the difference. That is, any other variables that might affect aggressive behavior must be controlled. There are two general categories of variables that researchers must consider: 1. Participant Variables These are characteristics such as age, gender, and intelligence that vary from one individual to another. Whenever an experiment compares different groups of participants (one group in treatment A and a different group in treatment B), researchers must ensure that participant variables do not differ from one group to another. For the experiment shown in Figure 1.5, for example, the researchers would like to conclude that the violence in the video game causes a change in the participants’ aggressive behavior. In the study, the participants in both conditions were 10-year-old boys. Suppose, however, that the participants in the nonviolent condition were primarily female and those in the violent condition were primarily male. In this case, there is an alternative explanation for the difference in aggression that exists between the two groups. Specifically, the difference between groups may have been caused by the amount of violence in the game, but it also is possible that the difference was caused by the participants’ gender (females are less aggressive than males). Whenever a research study allows more than one explanation for the results, the study is said to be confounded because it is impossible to reach an unambiguous conclusion. 2. Environmental Variables These are characteristics of the environment such as lighting, time of day, and weather conditions. A researcher must ensure that the individuals in treatment A are tested in the same environment as the individuals in treatment B. Using the video game violence experiment (see Figure 1.5) as an example, suppose that the individuals in the nonviolent condition were all tested in the morning and the individuals in the violent condition were all tested in the evening. Again, this would produce a confounded experiment because the researcher could not determine whether the differences in aggressive behavior were caused by the amount of violence or caused by the time of day. Researchers typically use three basic techniques to control other variables. First, the researcher could use random assignment, which means that each participant has an equal chance of being assigned to each of the treatment conditions. The goal of random assignment is to distribute the participant characteristics evenly between the two groups so that neither group is noticeably smarter (or older, or faster) than the other. Random assignment can also be used to control environmental variables. For example, participants could be assigned randomly for testing either in the morning or in the afternoon. A second technique SEctIon 1.2 | Data Structures, Research Methods, and Statistics 15 for controlling variables is to use matching to ensure equivalent groups or equivalent environments. For example, the researcher could match groups by ensuring that every group has exactly 60% females and 40% males. Finally, the researcher can control variables by holding them constant. For example, in the video game violence study discussed earlier (Polman et al., 2008), the researchers used only 10-year-old boys as participants (holding age and gender constant). In this case the researchers can be certain that one group is not noticeably older or has a larger proportion of females than the other. DEFInItIon In the experimental method, one variable is manipulated while another variable is observed and measured. To establish a cause-and-effect relationship between the two variables, an experiment attempts to control all other variables to prevent them from influencing the results. ■■Terminology in the Experimental Method Specific names are used for the two variables that are studied by the experimental method. The variable that is manipulated by the experimenter is called the independent variable. It can be identified as the treatment conditions to which participants are assigned. For the example in Figure 1.5, the amount of violence in the video game is the independent variable. The variable that is observed and measured to obtain scores within each condition is the dependent variable. For the example in Figure 1.5, the level of aggressive behavior is the dependent variable. DEFInItIon The independent variable is the variable that is manipulated by the researcher. In behavioral research, the independent variable usually consists of the two (or more) treatment conditions to which subjects are exposed. The independent variable consists of the antecedent conditions that were manipulated prior to observing the dependent variable. The dependent variable is the one that is observed to assess the effect of the treatment. Control Conditions in an Experiment An experimental study evaluates the relationship between two variables by manipulating one variable (the independent variable) and measuring one variable (the dependent variable). Note that in an experiment only one variable is actually measured. You should realize that this is different from a correlational study, in which both variables are measured and the data consist of two separate scores for each individual. Often an experiment will include a condition in which the participants do not receive any treatment. The scores from these individuals are then compared with scores from participants who do receive the treatment. The goal of this type of study is to demonstrate that the treatment has an effect by showing that the scores in the treatment condition are substantially different from the scores in the no-treatment condition. In this kind of research, the no-treatment condition is called the control condition, and the treatment condition is called the experimental condition. DEFInItIon Individuals in a control condition do not receive the experimental treatment. Instead, they either receive no treatment or they receive a neutral, placebo treatment. The purpose of a control condition is to provide a baseline for comparison with the experimental condition. Individuals in the experimental condition do receive the experimental treatment. 16 chaPtER 1 | Introduction to Statistics Note that the independent variable always consists of at least two values. (Something must have at least two different values before you can say that it is “variable.”) For the video game violence experiment (see Figure 1.5), the independent variable is the amount of violence in the video game. For an experiment with an experimental group and a control group, the independent variable is treatment versus no treatment. ■■Nonexperimental Methods: Nonequivalent Groups and Pre-Post Studies In informal conversation, there is a tendency for people to use the term experiment to refer to any kind of research study. You should realize, however, that the term only applies to studies that satisfy the specific requirements outlined earlier. In particular, a real experiment must include manipulation of an independent variable and rigorous control of other, extraneous variables. As a result, there are a number of other research designs that are not true experiments but still examine the relationship between variables by comparing groups of scores. Two examples are shown in Figure 1.6 and are discussed in the following paragraphs. This type of research study is classified as nonexperimental. The top part of Figure 1.6 shows an example of a nonequivalent groups study comparing boys and girls. Notice that this study involves comparing two groups of scores (like an experiment). However, the researcher has no ability to control which participants go into F i g u r E 1.6 (a) Two examples of nonexperimental studies that involve comparing two groups of scores. In (a) the study uses two preexisting groups (boys/ girls) and measures a dependent variable (verbal scores) in each group. In (b) the study uses time (before/after) to define the two groups and measures a dependent variable (depression) in each group. Variable #1: Subject gender (the quasi-independent variable) Not manipulated, but used to create two groups of subjects Variable #2: Verbal test scores (the dependent variable) Measured in each of the two groups Boys Girls 17 19 16 12 17 18 15 16 12 10 14 15 13 12 11 13 Any difference? (b) Variable #1: Time (the quasi-independent variable) Not manipulated, but used to create two groups of scores Variable #2: Depression scores (the dependent variable) Measured at each of the two different times Before Therapy After Therapy 17 19 16 12 17 18 15 16 12 10 14 15 13 12 11 13 Any difference? SEctIon 1.2 | Data Structures, Research Methods, and Statistics Correlational studies are also examples of nonexperimental research. In this section, however, we are discussing nonexperimental studies that compare two or more groups of scores. 17 which group—all the males must be in the boy group and all the females must be in the girl group. Because this type of research compares preexisting groups, the researcher cannot control the assignment of participants to groups and cannot ensure equivalent groups. Other examples of nonequivalent group studies include comparing 8-year-old children and 10-year-old children, people with an eating disorder and those with no disorder, and comparing children from a single-parent home and those from a two-parent home. Because it is impossible to use techniques like random assignment to control participant variables and ensure equivalent groups, this type of research is not a true experiment. The bottom part of Figure 1.6 shows an example of a pre–post study comparing depression scores before therapy and after therapy. The two groups of scores are obtained by measuring the same variable (depression) twice for each participant; once before therapy and again after therapy. In a pre-post study, however, the researcher has no control over the passage of time. The “before” scores are always measured earlier than the “after” scores. Although a difference between the two groups of scores may be caused by the treatment, it is always possible that the scores simply change as time goes by. For example, the depression scores may decrease over time in the same way that the symptoms of a cold disappear over time. In a pre–post study the researcher also has no control over other variables that change with time. For example, the weather could change from dark and gloomy before therapy to bright and sunny after therapy. In this case, the depression scores could improve because of the weather and not because of the therapy. Because the researcher cannot control the passage of time or other variables related to time, this study is not a true experiment. Terminology in Nonexperimental Research Although the two research studies shown in Figure 1.6 are not true experiments, you should notice that they produce the same kind of data that are found in an experiment (see Figure 1.5). In each case, one variable is used to create groups, and a second variable is measured to obtain scores within each group. In an experiment, the groups are created by manipulation of the independent variable, and the participants’ scores are the dependent variable. The same terminology is often used to identify the two variables in nonexperimental studies. That is, the variable that is used to create groups is the independent variable and the scores are the dependent variable. For example, the top part of Figure 1.6, gender (boy/girl), is the independent variable and the verbal test scores are the dependent variable. However, you should realize that gender (boy/girl) is not a true independent variable because it is not manipulated. For this reason, the “independent variable” in a nonexperimental study is often called a quasi-independent variable. DEFInItIon lE arn in g Ch ECk In a nonexperimental study, the “independent variable” that is used to create the different groups of scores is often called the quasi-independent variable. 1. In a correlational study, how many variables are measured for each individual and how many groups of scores are obtained? a. 1 variable and 1 group b. 1 variable and 2 groups c. 2 variables and 1 group d. 2 variables and 2 groups 18 chaPtER 1 | Introduction to Statistics 2. A research study comparing alcohol use for college students in the United States and Canada reports that more Canadian students drink but American students drink more (Kuo, Adlaf, Lee, Gliksman, Demers, and Wechsler, 2002). What research design did this study use? a. correlational b. experimental c. nonexperimental d. noncorrelational 3. Stephens, Atkins, and Kingston (2009) found that participants were able to tolerate more pain when they shouted their favorite swear words over and over than when they shouted neutral words. For this study, what is the independent variable? a. the amount of pain tolerated b. the participants who shouted swear words c. the participants who shouted neutral words d. the kind of word shouted by the participants an s wE r s 1.3 1. C, 2. C, 3. D Variables and Measurement LEARNING OBJECTIVEs 6. Explain why operational definitions are developed for constructs and identify the two components of an operational definition. 7. Describe discrete and continuous variables and identify examples of each. 8. Differentiate nominal, ordinal, interval, and ratio scales of measurement. ■■Constructs and Operational Definitions The scores that make up the data from a research study are the result of observing and measuring variables. For example, a researcher may finish a study with a set of IQ scores, personality scores, or reaction-time scores. In this section, we take a closer look at the variables that are being measured and the process of measurement. Some variables, such as height, weight, and eye color are well-defined, concrete entities that can be observed and measured directly. On the other hand, many variables studied by behavioral scientists are internal characteristics that people use to help describe and explain behavior. For example, we say that a student does well in school because he or she is intelligent. Or we say that someone is anxious in social situations, or that someone seems to be hungry. Variables like intelligence, anxiety, and hunger are called constructs, and because they are intangible and cannot be directly observed, they are often called hypothetical constructs. Although constructs such as intelligence are internal characteristics that cannot be directly observed, it is possible to observe and measure behaviors that are representative of the construct. For example, we cannot “see” intelligence but we can see examples of intelligent behavior. The external behaviors can then be used to create an operational definition for the construct. An operational definition defines a construct in terms of external SEctIon 1.3 | Variables and Measurement 19 behaviors that can be observed and measured. For example, your intelligence is measured and defined by your performance on an IQ test, or hunger can be measured and defined by the number of hours since last eating. DEFInItIon Constructs are internal attributes or characteristics that cannot be directly observed but are useful for describing and explaining behavior. An operational definition identifies a measurement procedure (a set of operations) for measuring an external behavior and uses the resulting measurements as a definition and a measurement of a hypothetical construct. Note that an operational definition has two components. First, it describes a set of operations for measuring a construct. Second, it defines the construct in terms of the resulting measurements. ■■Discrete and Continuous Variables The variables in a study can be characterized by the type of values that can be assigned to them. A discrete variable consists of separate, indivisible categories. For this type of variable, there are no intermediate values between two adjacent categories. Consider the values displayed when dice are rolled. Between neighboring values—for example, seven dots and eight dots—no other values can ever be observed. DEFInItIon A discrete variable consists of separate, indivisible categories. No values can exist between two neighboring categories. Discrete variables are commonly restricted to whole, countable numbers—for example, the number of children in a family or the number of students attending class. If you observe class attendance from day to day, you may count 18 students one day and 19 students the next day. However, it is impossible ever to observe a value between 18 and 19. A discrete variable may also consist of observations that differ qualitatively. For example, people can be classified by gender (male or female), by occupation (nurse, teacher, lawyer, etc.), and college students can by classified by academic major (art, biology, chemistry, etc.). In each case, the variable is discrete because it consists of separate, indivisible categories. On the other hand, many variables are not discrete. Variables such as time, height, and weight are not limited to a fixed set of separate, indivisible categories. You can measure time, for example, in hours, minutes, seconds, or fractions of seconds. These variables are called continuous because they can be divided into an infinite number of fractional parts. DEFInItIon For a continuous variable, there are an infinite number of possible values that fall between any two observed values. A continuous variable is divisible into an infinite number of fractional parts. Suppose, for example, that a researcher is measuring weights for a group of individuals participating in a diet study. Because weight is a continuous variable, it can be pictured as a continuous line (Figure 1.7). Note that there are an infinite number of possible points on 20 chaPtER 1 | Introduction to Statistics F i g u r E 1.7 149.6 When measuring weight to the nearest whole pound, 149.6 and 150.3 are assigned the value of 150 (top). Any value in the interval between 149.5 and 150.5 is given the value of 150. 150.3 149 151 152 151 152 150 149.5 149 148.5 150.5 150 149.5 150.5 151.5 152.5 Real limits the line without any gaps or separations between neighboring points. For any two different points on the line, it is always possible to find a third value that is between the two points. Two other factors apply to continuous variables: 1. When measuring a continuous variable, it should be very rare to obtain identical measurements for two different individuals. Because a continuous variable has an infinite number of possible values, it should be almost impossible for two people to have exactly the same score. If the data show a substantial number of tied scores, then you should suspect that the measurement procedure is very crude or that the variable is not really continuous. 2. When measuring a continuous variable, each measurement category is actually an interval that must be defined by boundaries. For example, two people who both claim to weigh 150 pounds are probably not exactly the same weight. However, they are both around 150 pounds. One person may actually weigh 149.6 and the other 150.3. Thus, a score of 150 is not a specific point on the scale but instead is an interval (see Figure 1.7). To differentiate a score of 150 from a score of 149 or 151, we must set up boundaries on the scale of measurement. These boundaries are called real limits and are positioned exactly halfway between adjacent scores. Thus, a score of X = 150 pounds is actually an interval bounded by a lower real limit of 149.5 at the bottom and an upper real limit of 150.5 at the top. Any individual whose weight falls between these real limits will be assigned a score of X = 150. DEFInItIon Real limits are the boundaries of intervals for scores that are represented on a continuous number line. The real limit separating two adjacent scores is located exactly halfway between the scores. Each score has two real limits. The upper real limit is at the top of the interval, and the lower real limit is at the bottom. The concept of real limits applies to any measurement of a continuous variable, even when the score categories are not whole numbers. For example, if you were measuring time to the nearest tenth of a second, the measurement categories would be 31.0, 31.1, 31.2, and so on. Each of these categories represents an interval on the scale that is bounded by real limits. For example, a score of X = 31.1 seconds indicates that the actual measurement is in an interval bounded by a lower real limit of 31.05 and an upper real limit of 31.15. Remember that the real limits are always halfway between adjacent categories. SEctIon 1.3 | Variables and Measurement Students often ask whether a value of exactly 150.5 should be assigned to the X = 150 interval or the X = 151 interval. The answer is that 150.5 is the boundary between the two intervals and is not necessarily in one or the other. Instead, the placement of 150.5 depends on the rule that you are using for rounding numbers. If you are rounding up, then 150.5 goes in the higher interval (X = 151) but if you are rounding down, then it goes in the lower interval (X = 150). 21 Later in this book, real limits are used for constructing graphs and for various calculations with continuous scales. For now, however, you should realize that real limits are a necessity whenever you make measurements of a continuous variable. Finally, we should warn you that the terms continuous and discrete apply to the variables that are being measured and not to the scores that are obtained from the measurement. For example, measuring people’s heights to the nearest inch produces scores of 60, 61, 62, and so on. Although the scores may appear to be discrete numbers, the underlying variable is continuous. One key to determining whether a variable is continuous or discrete is that a continuous variable can be divided into any number of fractional parts. Height can be measured to the nearest inch, the nearest 0.5 inch, or the nearest 0.1 inch. Similarly, a professor evaluating students’ knowledge could use a pass/fail system that classifies students into two broad categories. However, the professor could choose to use a 10-point quiz that divides student knowledge into 11 categories corre...