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Please see questions attachemnt (Total 8 questions needs to answer with details) and attaching textbook and syllabus Question 1: A program admissions coordinator wanted to know if the math placement test given to a certain cohort of students was an indicator on how well they would perform on their first exam.  She records the scores of their placement test and first exam below.  Is there a correlation between these measures? Placement Exam 1 11 49 27 71 7 60 33 85 28 80 23 75 20 61 48 100 35 71 26 81   Question 2: The scores below are performance scores for six subjects on Trial 1 and Trial 30 of a test of motor skills.  What test should you pick to measure if there is a relationship between trial 1 and trial 30 for the subjects?  What are the results of the statistical test (write up according to the handbook)? Trial 1      Trial 30                 3             21                 6             25                 2             20                 6             33                 5             23                 8             34 Question 3: Over the past decade, ABC Manufacturing has increased the number of safety inspections per month in hopes to decrease the number accidents on their production line.  Below is a list of how many inspections were conducted in a month and the corresponding number of accidents recorded.  Is there a correlation, and has this leadership decision been effective? Inspections Accidents 20 181 24 190 28 167 40 160 60 120 64 124 64 118 68 90 70 94 72 85 Question 4: The ACT standardized college admission exam was given in three rooms at a local testing site.  A complaint was filed that one of the testing conditions of each room were not the same and caused a difference of scores.  You have been asked to provide a statistical analysis to support or refute the claim.  Determine your hypotheses, run the appropriate test, and provide the appropriate conclusion. The scores from each of the three rooms were: Room A Room B Room C 21 17 19 16 24 16 14 29 22 17 21 15 12 17 16 19 30 13 34 22 29 14 16 17 26 12 27 36 19 27   Question 5: A marketing team conducted a survey of 300 people to see if a person’s age affected their brand choice.  People under the age of twenty-five made the following choices: Brand A (25), Brand B (42), Brand C (33).  People over 65 made the following selections: Brand A (20), Brand B (38), Brand C (42).  The group between those ages made these choices: Brand A (44), Brand B(40), Brand C (16).  Form your  hypotheses, run the appropriate test, and provide a conclusion to the marketing team.  Question 6: Five higher education institutions were ranked according to their cost and retention.  The data are listed below.  Which test would you use to determine if there is a correlation?  What are the results (written according to the handbook), and what do they mean? Cost Retention 1 3 2 2 3 4 4 1 5 5 Question 7: Five higher education institutions were ranked according to their cost and retention.  The data are listed below.  Which test would you use to determine if there is a correlation?  What are the results (written according to the handbook), and what do they mean? Cost Retention 1 2 2 3 3 5 4 1 5 4 Question 8: You have completed the DSRT 734 course.  It is time to apply your learning.  Respond to each of the following areas: Design a research question relevant to your specialty area.  Describe what type of data you would gather to answer the research question.  Determine which statistical test would be best to answer your research question.   What would you determine if the results were significant, based on your research question? Graduate School COURSE SYLLABUS Course Information DSRT734 - B04 Infer Stats in Decision-Making Summer 2021 Second Bi-Term Course Format: Online CRN: 30178 Class Time and Location: R 08:01 PM - 09:30 PM Instructor Information Name: Dr. Christopher Phillips Email: christopher [email protected] Phone: 606-451-6839 Office Location: Remote Office Hours/Preferred Contact Times: MTW 1-4 Course Description DSRT 734 − Inferential Statistics in Decision-making (3 hours) This course focuses on the descriptive and inferential statistics commonly used in organizational administration. Course activities include the application of statistical methods to research design, as well as an investigation of how these methods can contribute to school improvement. Specific concepts include measures of central tendency and variability, probability theory, estimation and significance, and correlation and regression. Course Objectives Upon completion of this course: Students will describe the essential characteristics of quantitative research. Students will apply inferential statistical tests on problems requiring parametric and non-parametric analyses. Students will identify potential research designs for various problems. Students will analyze statistical information in the professional literature on a topic related to their own research project. Learner Outcomes Students will analyze research and demonstrate professional communication skills.. Students will apply statistical tests to answer research questions. Students will write two article reviews, one practical connection assignment, complete three quizzes, one mid-term exam, and one final exam. Students will analyze statistical results. Students will learn the practical application of statistical analysis. Course Website Access to the course website is required via the iLearn portal on the University of the Cumberlands website: http://www.ucumberlands.edu/ilearn/ or h ttps://ucumberlands.blackboard.com/ Required Books and Resources Title: Publication Manual of the American Psychological Association ISBN: 9781433832178 Authors: American Psychological Association Publisher: American Psychological Association (APA) Publication Date: 2019-10-01 Title: Exploring Statistics ISBN: 9780996339223 Authors: Chris Spatz Publication Date: 2019-01-01 Course Required text can be found and purchased via the UC Barnes and Noble Bookstore: h ttps://cumber.bncollege.com/shop/cumberlands/page/find-textbooks Requirements and Policies A cademic Dishonesty Policy As a Christian liberal arts university committed to the pursuit of truth and understanding, any act of academic dishonesty is especially distressing and cannot be tolerated. In general, academic dishonesty involves the abuse and misuse of information or people to gain an undeserved academic advantage or evaluation. The common forms of academic dishonesty include: 1. cheating – using deception in the taking of tests or the preparation of written work, using unauthorized materials, copying another person’s work with or without consent, or assisting another in such activities; 2. lying – falsifying, fabricating, or forging information in either written or spoken presentations; 3. plagiarism – using the published writings, data, interpretations, or ideas of another without proper documentation. Episodes of academic dishonesty are reported as appropriate to the Vice President for Academic Affairs. The potential penalty for academic dishonesty includes 1) a failing grade on a particular assignment, 2) a failing grade for the entire course, 3) suspension or expulsion, or (4) revocation of a degree. Attendance Policy Course enrollment and participation will be monitored and verified for all students during the first two weeks of classes. Lack of participation during this time may jeopardize enrollment status. Each student is expected to meet course expectations by completing the coursework required each week. Active participation and staying abreast of the material is essential to success. Program specific attendance policies may still apply. P articip ation Policy Study after study has linked successful academic performance with good class participation. Those who assume positions of responsibility must “show up” in order to be effective. Therefore, students are expected to actively participate in an intelligent discussion of assigned topics in all areas (Discussion Board Activities, Synchronous Sessions, Forums, Shared Papers, etc.) to help process course material and/or to demonstrate an understanding of course content. Course Evaluations The course evaluation will be open during the last two weeks of the term. To access the evaluation (during that time), visit h ttps://uofcumberlands.campuslabs.com/eval-home/ and log-in using your UC credentials. A reminder email notification will be sent when the evaluation is available. We value your feedback. Every evaluation is confidential and anonymous. The anonymous results of the course evaluations are not available for faculty to see until after final grades are submitted. Your thoughtful responses guide future improvements for the course and programs. D isability Accommodations University of the Cumberlands accepts students with certified disabilities and provides reasonable accommodations for their certified needs in the classroom, in housing, in food service or in other areas. For accommodations to be awarded, a student must submit a completed Accommodations Application form and provide documentation of the disability to the Disability Services Coordinator (Keely Peterson, Boswell Campus Center, Student Services Office Suite, a ccommodations@ucumberlands.edu ). When all paperwork is on file, a meeting between the student and the Coordinator will be arranged to discuss possible accommodations before accommodations are formally approved. Students must then meet with the Coordinator at the beginning of each semester before any academic accommodations can be certified for that term. Certifications for other accommodations are normally reviewed annually. Academic Appeal Both undergraduate and graduate students have the right to challenge a grade. If discussions with the course instructor and department chair do not lead to a satisfactory conclusion, students may file a formal written appeal with the Vice President for Academic Affairs, who will forward the appeal to the chair of the Academic Appeals Committee. This formal written appeal must be filed by the end of the 4th week of classes in the next regular term following the term in which the course in question was taken. The Academic Appeals Committee then gathers information from the student, the instructor, and any other relevant parties. The Committee will deliver its recommendation on the complaint to the Vice President for Academic Affairs. After reviewing this recommendation and concurring or amending it, the Vice President for Academic Affairs will inform the student and instructor of the disposition of the complaint no later than the last day of classes of the term in which the complaint was filed. Records of all actions regarding academic grade appeals, including their final disposition, are maintained by the Vice President for Academic Affairs and the Academic Appeals Committee. ( U ndergraduate Catalog/Graduate Catalog ) Student Responsibilities Students should: Use University of the Cumberlands email system for all academic, administrative, and co-curricular communication between faculty, staff and peers. Check for email and class announcements using iLearn (primary) and University of the Cumberlands webmail (secondary) daily. Demonstrate Cumberlands Character in and outside the classroom per the University Mission & Vision Ensure you have consistent required technology for the course Participate in courses regularly to: Find announcements and updates Complete assignments on time. Keep in mind that all deadlines use Eastern Standard Time (EST). Engage in discussion Connect with fellow students and faculty Present written work in an academic and professional manner. Take examinations on the designated dates and times. Students should make arrangements with faculty before the designated date for any needed accommodations. Contact faculty or student success coordinator with questions or concerns. Course Policies The only authorized electronic means of academic, administrative, and co- curricular communication between University of the Cumberlands and its students is through the UCumberlands email system (i.e. Webmail). Each student is responsible for monitoring his/her University email account frequently. This is the primary email account used to correspond with you directly by the University; imperative program information is sent to this email account specifically from campus and program office. Students should check for e-mail and class announcements using iLearn (primary) and University of the Cumberlands webmail (secondary). Students are expected to find out class assignments for missed classes and make up missed work. Students are expected to find out if any changes have been made in the class or assignment schedule. All assignments, unless otherwise instructed, should be submitted in APA format. Written work must be presented in a professional manner. Work that is not submitted in a professional manner will not be evaluated and will be returned as unacceptable. There is a craft to writing. Spelling, grammar, punctuation, and diction (word usage) are all tools of that craft. Writing at the collegiate level will show careful attention to these elements of craft. Work that does not exhibit care with regard to these elements will be considered as inadequate for college writing and graded accordingly. Students are expected to take the examinations on the designated dates. If you are unable to take the exam on the scheduled date and know in advance, you are to make arrangements with your professor before the designated date. If you miss the exam, you must have a legitimate reason as determined by your professor. Recognizing that a large part of professional life is meeting deadlines, it is necessary to develop time management and organizational skills. Failure to meet the course deadlines may result in penalties. Keep in mind that all deadlines are set using Eastern Standard Time (EST). Late assignments will NOT be accepted. Course Activities and Experiences Graded work will receive a numeric score reflecting the quality of performance. There are 500 points possible. Students will be evaluated on: 1. Three Quizzes (50 points each) 2. Two Article Reviews (25 points each) Written Assignment where students evaluate one quantitative and one qualitative research article (One page each). 3. Practical Connection Assignment (50 points total) Written Assignment where students will reflect on course concepts and their practical connection to a working environment. 4. Comprehensive Mid-Term Exam (125 points total) 5. Comprehensive Final Exam (125 points total) Links to Support UC Academic Catalog: h ttps://www.ucumberlands.edu/academics/academic-catalog UC Student Handbook: h ttps://www.ucumberlands.edu/student-handbook Academic Resources & Writing Center: w ww.ucumberlands.edu/learningcommons Library: h ttp://www.ucumberlands.edu/library/ Bookstore: h ttps://cumber.bncollege.com/shop/cumberlands/home About University of the Cumberlands: h ttps://www.ucumberlands.edu/about/presidents-welcome Instructions for Accessing, Downloading, and Activating Office 365 Pro Plus (free for UC Students): h ttps://helpdesk.ucumberlands.edu/support/solutions/articles/7000045435 Course Evaluation Graded work will receive a numeric score reflecting the quality of performance. There are 500 points possible. Students will be evaluated on: 1. Quizzes (Three quizzes at 50 points each) 2. Two article reviews (one page each) evaluating one quantitative and one qualitative research article (50 points total) 3. Practical Connection Assignment - Written Assignment where students will reflect on course concepts and their practical connection to a working environment. (50 points total) 4. Comprehensive Mid-Term Exam (125 points total) 5. Comprehensive Final Exam (125 points total) Grading Scale A: 90\% - 100\% B: 80\% - 89\% C: 70\% - 79\% F: 69\% or Below Course Schedule Week 1 (June 28 - July 4) Blackboard July 1 @ 8 pm Introduction & Descriptive Statistics Chapter 1: Introduction Chapter 2: Frequency Distributions and Graphs Chapter 3: Central Tendency Chapter 4: Variability Chapter 5: Other Descriptive Statistics “The best way for you to acquire all these skills (especially the part about telling the story) is to engage statistics.” (Spatz, 2019, p. 6) Required: Complete Introduction Discussion Board Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. *Failing to Participate in Week 1 may result in being dropped from the course. Week 2 (July 5-11) Blackboard July 8 @ 8 pm Regression, Distributions, & Samples Chapter 6: Correlation and Regression Chapter 7: Theoretical Distributions Including the Normal Distribution Chapter 8: Samples “Statistics is a powerful method for getting answers from data ...” (Spatz, 2019, p. 5) Required: Quiz 1 (Chapters 1-5) - due by July 11 @ 11:5 PM (EST) Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Week 3 (July 12-18) Blackboard July 15 Paired and Independent Samples t-tests Chapter 9: Effect Size and NHST: One-Sample Designs Required: Quiz 2 (Chapters 6-10) - due by July 18,2021 @ 11:59 PM (EST) @ 8 pm Chapter 10: Effect Size, Confidence Intervals, and NHST: Two-Sample Designs “Statistics is just one tool that researchers use, but it is often an essential tool.” (Spatz, 2019, p. 9) ANOVA Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Week 4 (July 19-25) Blackboard July 22 @ 8 pm Week 5 (July 26 - Aug 1) Blackboard July 29 @ 8 pm Chapter 11: Analysis of Variance: Independent Samples Chapter 12 - Analysis of Variance: Repeated Measures Chapter 13 - Analysis of Variance: Factorial Design “A very common task of most human beings can be described as trying to understand. Statistics has helped many in their search for better understanding ...” (Spatz, 2019, p. 21) Chi-Square Tests Chapter 14: Chi-Square “Your active participation is necessary if you are to learn statistics.” (Spatz, 2019, p. 22) Required: Comprehensive Mid- Term Exam - due by July 25 @ 11:59 PM (EST) Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Required: Article Reviews - due by Aug 1 @ 11:59 PM (EST) Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Week 6 (Aug 2-8) Blackboard Aug 5 @ 8 pm Spearman’s rs Chapter 15: Non-Parametric Tests “Being able to calculate a statistic is almost worthless if you cannot explain in English what it means. Writing reveals how thoroughly you understand.” (Spatz, 2019, p. 23) Required: Quiz 3 (Chapters 14- 15) - due by Aug 8 @ 11:59 PM (EST) Recommended: Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Final Exam Review and Writing Research Questions Chapter 16: Choosing Tests and Writing Interpretations “Calculating numbers or drawing graphs is a part of almost every statistical problem, but unless you can tell the story of what the numbers and pictures mean, you won’t find statistics worthwhile.” (Spatz, 2019, p. 47) Required: Week 7 (Aug 9-15) Practical Connection Assignment – due by Aug 15 @ 11:59 PM (EST) Recommended: Blackboard Aug 12 @ 8 pm Additional Excel/Stats practice Actively engage practice problems in text for each chapter. Week 8 (Aug 16-19) Comprehensive Final Exam (All Lessons) No Live Class in Collaborate Required: Comprehensive Final Exam – due by Aug 19 2021 @ 11:59 PM (EST NO EXCEPTIONS Recommended: Review Excel/Stats practices. “This phrase [Exploring Data] is a Review practice problems in text for each chapter. reminder to approach a data set with the attitude of an explorer, an attitude of What can I find here?” (Spatz, 2019, p. 28) Syllabus Disclaimer This syllabus contains important information critical to your success in this course. It includes guidelines for this course and the instructor’s current expectations about content, schedule, and requirements necessary for each student to achieve the best educational results. While you must review and become familiar with the contents of this syllabus, the instructor reserves the right to make adjustments or change in the syllabus from time to time. Any changes to the syllabus will be discussed with the students. Exploring Statistics Tales of Distributions 12th Edition Chris Spatz Outcrop Publishers Conway, Arkansas Exploring Statistics: Tales of Distributions 12th Edition Chris Spatz Cover design: Grace Oxley Answer Key: Jill Schmidlkofer Webmaster & Ebook: Fingertek Web Design, Tina Haggard Managers: Justin Murdock, Kevin Spatz Copyright © 2019 by Outcrop Publishers, LLC All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, contact [email protected] or write to the publisher at the address below. Outcrop Publishers 615 Davis Street Conway, AR 72034 Email: [email protected] Website: outcroppublishers.com Library of Congress Control Number: [Applied for] ISBN-13 (hardcover): 978-0-9963392-2-3 ISBN-13 (ebook): 978-0-9963392-3-0 ISBN-13 (study guide): 978-0-9963392-4-7 Examination copies are provided to academics and professionals to consider for adoption as a course textbook. Examination copies may not be sold or transferred to a third party. If you adopt this textbook, please accept it as your complimentary desk copy. Ordering information: Students and professors – visit exploringstatistics.com Bookstores – email [email protected] Photo Credits – Chapter 1 Karl Pearson – Courtesy of Wellcomeimages.org Ronald A. Fisher – R.A. Fisher portrait, 0006973, Special Collections Research Center, North Carolina State University Libraries, Raleigh, North Carolina Jerzy Neyman – Paul R. Halmos Photograph Collection, e_ph 0223_01, Dolph Briscoe Center for American History, The University of Texas at Austin Jacob Cohen – New York University Archives, Records of the NYU Photo Bureau Printed in the United States of America by Walsworth ® 2 3 4 5 6 7 24 23 22 21 20 Online study guide available at http://exploringstatistics.com/studyguide.php http://exploringstatistics.com/studyguide.php mailto:[email protected] http://outcroppublishers.com http://Wellcomeimages.org mailto:[email protected] http://exploringstatistics.com v About The Author Chris Spatz is at Hendrix College where he twice served as chair of the Psychology Department. Dr. Spatz’s undergraduate education was at Hendrix, and his PhD in experimental psychology is from Tulane University in New Orleans. He subsequently completed postdoctoral fellowships in animal behavior at the University of California, Berkeley, and the University of Michigan. Before returning to Hendrix to teach, Spatz held positions at The University of the South and the University of Arkansas at Monticello. Spatz served as a reviewer for the journal Teaching of Psychology for more than 20 years. He co-authored a research methods textbook, wrote several chapters for edited books, and was a section editor for the Encyclopedia of Statistics in Behavioral Science. In addition to writing and publishing, Dr. Spatz enjoys the outdoors, especially canoeing, camping, and gardening. He swims several times a week (mode = 3). Spatz has been an opponent of high textbook prices for years, and he is happy to be part of a new wave of authors who provide high-quality textbooks to students at affordable prices. About The Author vi Dedication With love and affection, this textbook is dedicated to Thea Siria Spatz, Ed.D., CHES vii Brief Contents Brief Contents Preface xiv 1 Introduction 1 2 Exploring Data: Frequency Distributions and Graphs 29 3 Exploring Data: Central Tendency 45 4 Exploring Data: Variability 59 5 Other Descriptive Statistics 77 6 Correlation and Regression 94 7 Theoretical Distributions Including the Normal Distribution 127 8 Samples, Sampling Distributions, and Confidence Intervals 150 9 Effect Size and NHST: One-Sample Designs 175 10 Effect Size, Confidence Intervals, and NHST: Two-Sample Designs 200 11 Analysis of Variance: Independent Samples 231 12 Analysis of Variance: Repeated Measures 259 13 Analysis of Variance: Factorial Design 271 14 Chi Square Tests 303 15 More Nonparametric Tests 328 16 Choosing Tests and Writing Interpretations 356 Appendixes A Getting Started 371 B Grouped Frequency Distributions and Central Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414 References 466 Index 472 viii Preface xiv chapter 1 Introduction 1 Disciplines That Use Quantitative Data 5 What Do You Mean, “Statistics”? 6 Statistics: A Dynamic Discipline 8 Some Terminology 9 Problems and Answers 12 Scales of Measurement 13 Statistics and Experimental Design 16 Experimental Design Variables 17 Statistics and Philosophy 20 Statistics: Then and Now 21 How to Analyze a Data Set 22 Helpful Features of This Book 22 Computers, Calculators, and Pencils 24 Concluding Thoughts 25 Key Terms 27 Transition Passage to Descriptive Statistics 28 chapter 2 Exploring Data: Frequency Distributions and Graphs 29 Simple Frequency Distributions 31 Grouped Frequency Distributions 33 Graphs of Frequency Distributions 35 Describing Distributions 39 Contents Contents ix The Line Graph 41 More on Graphics 42 A Moment to Reflect 43 Key Terms 44 chapter 3 Exploring Data: Central Tendency 45 Measures of Central Tendency 46 Finding Central Tendency of Simple Frequency Distributions 49 When to Use the Mean, Median, and Mode 52 Determining Skewness From the Mean and Median 54 The Weighted Mean 55 Estimating Answers 56 Key Terms 58 chapter 4 Exploring Data: Variability 59 Range 61 Interquartile Range 61 Standard Deviation 63 Standard Deviation as a Descriptive Index of Variability 64 ŝ as an Estimate of σ 69 Variance 73 Statistical Software Programs 74 Key Terms 76 chapter 5 Other Descriptive Statistics 77 Describing Individual Scores 78 Boxplots 82 Effect Size Index 86 The Descriptive Statistics Report 89 Key Terms 92 Transition Passage to Bivariate Statistics 93 chapter 6 Correlation and Regression 94 Bivariate Distributions 96 Positive Correlation 96 Negative Correlation 99 Zero Correlation 101 Correlation Coefficient 102 Scatterplots 106 Contents x Interpretations of r 106 Uses of r 110 Strong Relationships but Low Correlation Coefficients 112 Other Kinds of Correlation Coefficients 115 Linear Regression 116 The Regression Equation 117 Key Terms 124 What Would You Recommend? Chapters 2-6 125 Transition Passage to Inferential Statistics 126 chapter 7 Theoretical Distributions Including the Normal Distribution 127 Probability 128 A Rectangular Distribution 129 A Binomial Distribution 130 Comparison of Theoretical and Empirical Distributions 131 The Normal Distribution 132 Comparison of Theoretical and Empirical Answers 146 Other Theoretical Distributions 146 Key Terms 147 Transition Passage to the Analysis of Data From Experiments 149 chapter 8 Samples, Sampling Distributions, and Confidence Intervals 150 Random Samples 152 Biased Samples 155 Research Samples 156 Sampling Distributions 157 Sampling Distribution of the Mean 157 Central Limit Theorem 159 Constructing a Sampling Distribution When σ Is Not Available 164 The t Distribution 165 Confidence Interval About a Population Mean 168 Categories of Inferential Statistics 172 Key Terms 173 Contents xi Transition Passage to Null Hypothesis Significance Testing 174 chapter 9 Effect Size and NHST: One-Sample Designs 175 Effect Size Index 176 The Logic of Null Hypothesis Significance Testing (NHST) 179 Using the t Distribution for Null Hypothesis Significance Testing 182 A Problem and the Accepted Solution 184 The One-Sample t Test 186 An Analysis of Possible Mistakes 188 The Meaning of p in p < .05 191 One-Tailed and Two-Tailed Tests 192 Other Sampling Distributions 195 Using the t Distribution to Test the Significance of a Correlation Coefficient 195 t Distribution Background 197 Why .05? 198 Key Terms 199 chapter 10 Effect Size, Confidence Intervals, and NHST: Two-Sample Designs 200 A Short Lesson on How to Design an Experiment 201 Two Designs: Paired Samples and Independent Samples 202 Degrees of Freedom 206 Paired-Samples Design 208 Independent-Samples Design 212 The NHST Approach 217 Statistical Significance and Importance 222 Reaching Correct Conclusions 222 Statistical Power 225 Key Terms 228 What Would You Recommend? Chapters 7-10 229 Transition Passage to More Complex Designs 230 Contents xii Contents chapter 11 Analysis of Variance: Independent Samples 231 Rationale of ANOVA 233 More New Terms 240 Sums of Squares 240 Mean Squares and Degrees of Freedom 245 Calculation and Interpretation of F Values Using the F Distribution 246 Schedules of Reinforcement—A Lesson in Persistence 248 Comparisons Among Means 250 Assumptions of the Analysis of Variance 254 Random Assignment 254 Effect Size Indexes and Power 255 Key Terms 258 chapter 12 Analysis of Variance: Repeated Measures 259 A Data Set 260 Repeated-Measures ANOVA: The Rationale 261 An Example Problem 262 Tukey HSD Tests 265 Type I and Type II Errors 266 Some Behind-the-Scenes Information About Repeated-Measures ANOVA 267 Key Terms 270 chapter 13 Analysis of Variance: Factorial Design 271 Factorial Design 272 Main Effects and Interaction 276 A Simple Example of a Factorial Design 282 Analysis of a 2 × 3 Design 291 Comparing Levels Within a Factor—Tukey HSD Tests 297 Effect Size Indexes for Factorial ANOVA 299 Restrictions and Limitations 299 Key Terms 301 Transition Passage to Nonparametric Statistics 302 chapter 14 Chi Square Tests 303 The Chi Square Distribution and the Chi Square Test 305 Chi Square as a Test of Independence 307 Shortcut for Any 2 × 2 Table 310 Effect Size Indexes for 2 × 2 Tables 310 Chi Square as a Test for Goodness of Fit 314 xiii Contents Chi Square With More Than One Degree of Freedom 316 Small Expected Frequencies 321 When You May Use Chi Square 324 Key Terms 327 chapter 15 More Nonparametric Tests 328 The Rationale of Nonparametric Tests 329 Comparison of Nonparametric to Parametric Tests 330 Mann-Whitney U Test 332 Wilcoxon Signed-Rank T Test 339 Wilcoxon-Wilcox Multiple-Comparisons Test 344 Correlation of Ranked Data 348 Key Terms 353 What Would You Recommend? Chapters 11-15 353 chapter 16 Choosing Tests and Writing Interpretations 356 A Review 356 My (Almost) Final Word 357 Future Steps 358 Choosing Tests and Writing Interpretations 359 Key Term 368 Appendixes A Getting Started 371 B Grouped Frequency Distributions and Central Tendency 376 C Tables 380 D Glossary of Words 401 E Glossary of Symbols 405 F Glossary of Formulas 407 G Answers to Problems 414 References 466 Index 472 xiv Preface Exploring Statistics: Tales of Distributions (12th edition) is a textbook for a one-term statistics course in the social or behavioral sciences, education, or an allied health/nursing field. Its focus is conceptualization, understanding, and interpretation, rather than computation. Designed to be comprehensible and complete for students who take only one statistics course, it also includes elements that prepare students for additional statistics courses. For example, basic experimental design terms such as independent and dependent variables are explained so students can be expected to write fairly complete interpretations of their analyses. In many places, the student is invited to stop and think or do a thought exercise. Some problems ask the student to decide which statistical technique is appropriate. In sum, this book’s approach is in tune with instructors who emphasize critical thinking in their course. This textbook has been remarkably successful for more than 40 years. Students, professors, and reviewers have praised it. A common refrain is that the book has a conversational, narrative style that is engaging, especially for a statistics text. Other features that distinguish this textbook from others include the following: • Data sets are approached with an attitude of exploration. • Changes in statistical practice over the years are acknowledged, especially the recent emphasis on effect sizes and confidence intervals. • Criticism of null hypothesis significance testing (NHST) is explained. • Examples and problems represent a variety of disciplines and everyday life. • Most problems are based on actual studies rather than fabricated scenarios. • Interpretation is emphasized throughout. • Problems are interspersed within a chapter, not grouped at the end. • Answers to all problems are included. • Answers are comprehensively explained—over 50 pages of detail. • A final chapter, Choosing Tests and Writing Interpretations, requires active responses to comprehensive questions. Preface Even if our statistical appetite is far from keen, we all of us should like to know enough to understand, or to withstand, the statistics that are constantly being thrown at us in print or conversation—much of it pretty bad statistics. The only cure for bad statistics is apparently more and better statistics. All in all, it certainly appears that the rudiments of sound statistical sense are coming to be an essential of a liberal education. – Robert Sessions Woodworth xv Preface • Effect size indexes are treated as important descriptive statistics, not add-ons to NHST. • Important words and phrases are defined in the margin when they first occur. • Objectives, which open each chapter, serve first for orientation and later as review items. • Key Terms are identified for each chapter. • Clues to the Future alert students to concepts that come up again. • Error Detection boxes tell ways to detect mistakes or prevent them. • Transition Passages alert students to a change in focus in chapters that follow. • Comprehensive Problems encompass all (or most) of the techniques in a chapter. • What Would You Recommend? problems require choices from among techniques in several chapters. For this 12th edition, I increased the emphasis on effect sizes and confidence intervals, moving them to the front of Chapter 9 and Chapter 10. The controversy over NHST is addressed more thoroughly. Power gets additional attention. Of course, examples and problems based on contemporary data are updated, and there are a few new problems. In addition, a helpful Study Guide to Accompany Exploring Statistics (12th edition) was written by Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, all of Hendrix College. The study guide is available online at exploringstatistics.com. Students who engage in this book and their course can expect to: • Solve statistical problems • Understand and explain statistical reasoning • Choose appropriate statistical techniques for common research designs • Write explanations that are congruent with statistical analyses After many editions with a conventional publisher, Exploring Statistics: Tales of Distributions is now published by Outcrop Publishers. As a result, the price of the print edition is about one-fourth that of the 10th edition. Nevertheless, the authorship and quality of earlier editions continue as before. xvi Preface Acknowledgments The person I acknowledge first is the person who most deserves acknowledgment. And for the 11th and 12th editions, she is especially deserving. This book and its accompanying publishing company, Outcrop Publishers, would not exist except for Thea Siria Spatz, encourager, supporter, proofreader, and cheer captain. This edition, like all its predecessors, is dedicated to her. Kevin Spatz, manager of Outcrop Publishers, directed the distribution of the 11th edition, advised, week by week, and suggested the cover design for the 12th edition. Justin Murdock now serves as manager, continuing the tradition that Kevin started. Tina Haggard of Fingertek Web Design created the book’s website, the text’s ebook, and the online study guide. She provided advice and solutions for many problems. Thanks to Jill Schmidlkofer, who edited the extensive answer section again for this edition. Emily Jones Spatz created new drawings for the text. I’m particularly grateful to Grace Oxley for a cover design that conveys exploration, and to Liann Lech, who copyedited for clarity and consistency. Walsworth® turned a messy collection of files into a handsome book—thank you Nathan Stufflebean and Dennis Paalhar. Others who were instrumental in this edition or its predecessors include Jon Arms, Ellen Bruce, Mary Kay Dunaway, Bob Eslinger, James O. Johnston, Roger E. Kirk, Rob Nichols, Jennifer Peszka, Mark Spatz, and Selene Spatz. I am especially grateful to Hendrix College and my Hendrix colleagues for their support over many years, and in particular, to Lindsay Kennedy, Jennifer Peszka, and Leslie Zorwick, who wrote the study guide that accompanies the text. This textbook has benefited from perceptive reviews and significant suggestions by some 90 statistics teachers over the years. For this 12th edition, I particularly thank Jessica Alexander, Centenary College Lindsay Kennedy, Hendrix College Se-Kang Kim, Fordham University Roger E. Kirk, Baylor University Kristi Lekies, The Ohio State University Jennifer Peszka, Hendrix College Robert Rosenthal, University of California, Riverside I’ve always had a touch of the teacher in me—as an older sibling, a parent, a professor, and now a grandfather. Education is a first-class task, in my opinion. I hope this book conveys my enthusiasm for it. (By the way, if you are a student who is so thorough as to read even the acknowledgments, you should know that I included phrases and examples in a number of places that reward your kind of diligence.) If you find errors in this book, please report them to me at [email protected] I will post corrections at the book’s website: exploringstatistics.com. Introduction CHAPTER 1 O B J E C T I V E S F O R C H A P T E R 1 After studying the text and working the problems in this chapter, you should be able to: 1. Distinguish between descriptive and inferential statistics 2. Define population, sample, parameter, statistic, and variable as they are used in statistics 3. Distinguish between quantitative and categorical variables 4. Distinguish between continuous and discrete variables 5. Identify the lower and upper limits of a continuous variable 6. Identify four scales of measurement and distinguish among them 7. Distinguish between statistics and experimental design 8. Define independent variable, dependent variable, and extraneous variable and identify them in experiments 9. Describe statistics’ place in epistemology 10. List actions to take to analyze a data set 11. Identify a few events in the history of statistics WE BEGIN OUR exploration of statistics with a trip to London. The year is 1900. Walking into an office at University College London, we meet a tall, well-dressed man about 40 years old. He is Karl Pearson, Professor of Applied Mathematics and Mechanics. I ask him to tell us a little about himself and why he is an important person. He seems authoritative, glad to talk about himself. As a young man, he says, he wrote essays, a play, and a novel, and he also worked for women’s suffrage. These days, he is excited about this new branch of biology called genetics. He says he supervises lots of data gathering. 1 Karl Pearson 2 Chapter 1 Pearson, warming to our group, lectures us about the major problem in science—there is no agreement on how to decide among competing theories. Fortunately, he just published a new statistical method that provides an objective way to decide among competing theories, regardless of the discipline. The method is called chi square.1 Pearson says, “Now, arguments will be much fewer. Gather a thousand data points and calculate a chi square test. The result gives everyone an objective way to determine whether or not the data fit the theory.” Exploration Notes from a student: Exploration off to good start. Hit on a nice, easy-to- remember date to start with, visited a founder of statistics, and had a statistic called chi square described as a big deal. Our next stop is Rothamsted Experiment Station just north of London. Now the year is 1925. There are fields all around the agricultural research facility, each divided into many smaller plots. The growth in the fields seems quite variable. Arriving at the office, the atmosphere is congenial. The staff is having tea. There are two topics—a new baby and a new book. We get introduced to Ronald Fisher, the chief statistician. Fisher is a small man with thick glasses and red hair. He tells us about his new child2 and then motions to a book on the table. Sneaking a peek, we read the title: Statistical Methods for Research Workers. Fisher becomes focused on his book, holding forth in an authoritative way. He says the book explains how to conduct experiments and that an experiment is just a comparison of two or more conditions. He tells us we don’t need a thousand data points. He says that small samples, randomly selected, are the way for science to progress. “With an experiment and my technique of analysis of variance,” he exclaims, “you can determine why that field out there”—here he waves toward the window—“is so variable. We can find out what makes some plots lush and some mimsy.” Analysis of variance,3 he says, works in any discipline, not just agriculture. Exploration Notes: Looks like statistics had some controversy in it.4 Also looks like progress. Statistics is used for experiments, too, and not just for testing theories. And Fisher says experiments can be used to compare anything. If that’s right, I can use statistics no matter what I major in. 1 Chi square, which is explained in this book in Chapter 14, has been called one of the 20 most important inventions in the 20th century (Hacking, 1984). 2 (in what will become a family with eight children). 3 explained in Chapters 11-13 4 The slight sniping I’ve built into this story is just a hint of the strong animosity between Fisher and Pearson. Ronald A. Fisher 3 Introduction Next we go to Poland to visit Jerzy Neyman at his office at the University of Warsaw. It is 1933. As we walk in, he smiles, seems happy we’ve arrived, and makes us feel completely welcome. Motioning to an envelope on his desk, he tells us it holds a manuscript that he and Egon Pearson5 wrote. “The problem with Fisher’s analysis of variance test is that it focuses exclusively on finding a difference between groups. Suppose the statistical test doesn’t detect a difference. Does that prove there is no difference? No, of course not. It may be that the test was just not sensitive enough to detect the difference. Right?” At his question, a few of us nod in agreement. Seeing uncertainty, he notes, “Maybe a larger sample is needed to find the difference, you see? Anyway, what we’ve done is expand statistics to cover not just finding a difference, but also what it means when the test doesn’t find a difference. Our approach is what you people in your time will call null hypothesis significance testing.” Exploration Notes: Statistics seems like a work in progress. Changing. Now it is not just about finding a difference but also about what it means not to find a difference. Also, looks like null hypothesis significance testing is a phrase that might turn up on tests. Our next trip is to libraries, say, anytime between 1940 and 2000. For this exploration, the task is to examine articles in professional journals published in various disciplines. The disciplines include anthropology, biology, chemistry, defense strategy, education, forestry, geology, health, immunology, jurisprudence, manufacturing, medicine, neurology, ophthalmology, political science, psychology, sociology, zoology, and others. I’m sure you get the idea—the whole range of disciplines that use quantitative measures in their research. What this exploration produces is the discovery that all of these disciplines rely on a data analysis technique called null hypothesis significance testing (NHST).6 Many different statistical tests are employed. However, for all the tests in all the disciplines, the phrase, “p < .05” turns up frequently. Exploration Notes: It seems that all that earlier controversy has subsided and scientists in all sorts of disciplines have agreed that NHST is the way to analyze quantitative data. All of them seem to think that if there is a comparison to be made, applying NHST is a necessary step to get correct conclusions. All of them use “p < .05,” so I’ll have to be sure to find out exactly what that means. 5 Egon Pearson was Karl Pearson’s son. 6 Null hypothesis significance testing is first explained in Chapters 9 and 10. Jerzy Neyman 4 Chapter 1 Our next excursion is a 1962 visit with Jacob Cohen at New York University in New York City. He is holding his article about studies published in the Journal of Abnormal and Social Psychology, a leading psychology journal. He tells us that the NHST technique has problems. Also, he says we should be calculating an effect size statistic, which will show whether the differences observed in our experiments are large or small. Exploration Notes: The idea of an effect size index makes a lot of sense. Just knowing there is a difference isn’t enough. How big is the difference? Wonder what “problems with NHST” is all about. Back to the library for a final excursion to check out recent events. We come across a 2014 article by Geoff Cumming on the “new statistics.” We find things like, “avoid NHST and use better techniques” (p. 26) and “we should not trust any p value” (p. 13). This seems like awfully strong advice. Are researchers taking this advice? Looking through more of today’s research in journals in several fields, we find that most statistical analyses use NHST and there are many instances of “p < .05.” Exploration Notes, Conclusion: These days, it looks like statistics is in transition again. There’s a lot of controversy out there about how to analyze data from experiments. The NHST approach is still very common, though, so it’s clear I must learn it. But I want to be prepared for changes. I hope knowing NHST will be helpful for the future.7 Welcome to statistics at a time when the discipline is once again in transition. A well- established tradition (null hypothesis significance testing) has been in place for almost a century but is now under attack. New ways of thinking about data analysis are emerging, and along with them, a collection of statistics that do not include the traditional NHST approach. As for the immediate future, though, NHST remains the method most widely used by researchers in many fields. In addition, much of the thinking required for NHST is required for other approaches. Our exploration tour is over, so I’ll quit supplying notes; they are your responsibility now. As your own experience probably shows, making up your own summary notes improves retention of what you read. In addition, I have a suggestion. Adopt a mindset that thinks growth. A student with a growth mindset expects to learn new things. When challenges arise, as they 7 Not only helpful, but necessary, I would say. Jacob Cohen 5 Introduction Disciplines that Use Quantitative Data inevitably do, acknowledge them and figure out how to meet the challenge. A growth mindset treats ability as something to be developed (see Dweck, 2016). If you engage yourself in this course, you can expect to use what you learn for the rest of your life. The main title of this book is “Exploring Statistics.” Exploring conveys the idea of uncovering something that was not apparent before. An attitude of searching, wondering, checking, and so forth is what I want to encourage. (Those who object to traditional NHST procedures are driven by this exploration motivation.) As for this book’s subtitle, “Tales of Distributions,” I’ll have more to say about it as we go along. Which disciplines use quantitative data? The list is long and more variable than the list I gave earlier. The examples and problems in this textbook, however, come from psychology, biology, sociology, education, medicine, politics, business, economics, forestry, and everyday life. Statistics is a …