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The future of education. openstax.org Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter1: Sampling and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Definitions of Statistics, Probability, and Key Terms . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Data, Sampling, and Variation in Data and Sampling . . . . . . . . . . . . . . . . . . . . . 10 1.3 Frequency, Frequency Tables, and Levels of Measurement . . . . . . . . . . . . . . . . . . 26 1.4 Experimental Design and Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.5 Data Collection Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6 Sampling Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter2: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs . . . . . . . . . . . . . . 68 2.2 Histograms, Frequency Polygons, and Time Series Graphs . . . . . . . . . . . . . . . . . . 77 2.3 Measures of the Location of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4 Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.5 Measures of the Center of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.6 Skewness and the Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.7 Measures of the Spread of the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.8 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Chapter3: Probability Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.2 Independent and Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.3 Two Basic Rules of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.4 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3.5 Tree and Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 3.6 Probability Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Chapter4: Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable . . . . . . . . . . . 244 4.2 Mean or Expected Value and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 247 4.3 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 4.4 Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 4.5 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.6 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 4.7 Discrete Distribution (Playing Card Experiment) . . . . . . . . . . . . . . . . . . . . . . . . 271 4.8 Discrete Distribution (Lucky Dice Experiment) . . . . . . . . . . . . . . . . . . . . . . . . . 274 Chapter5: Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 5.1 Continuous Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 5.2 The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.3 The Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 5.4 Continuous Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Chapter6: The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 6.1 The Standard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 6.2 Using the Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 6.3 Normal Distribution (Lap Times) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 6.4 Normal Distribution (Pinkie Length) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Chapter7: The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 7.1 The Central Limit Theorem for Sample Means (Averages) . . . . . . . . . . . . . . . . . . 400 7.2 The Central Limit Theorem for Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 7.3 Using the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 7.4 Central Limit Theorem (Pocket Change) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 7.5 Central Limit Theorem (Cookie Recipes) . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 Chapter8: Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 8.1 A Single Population Mean using the Normal Distribution . . . . . . . . . . . . . . . . . . . 445 8.2 A Single Population Mean using the Student t Distribution . . . . . . . . . . . . . . . . . . 456 8.3 A Population Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 8.4 Confidence Interval (Home Costs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 8.5 Confidence Interval (Place of Birth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 8.6 Confidence Interval (Women's Heights) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Chapter9: Hypothesis Testing with One Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 9.1 Null and Alternative Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 9.2 Outcomes and the Type I and Type II Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 508 9.3 Distribution Needed for Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 510 9.4 Rare Events, the Sample, Decision and Conclusion . . . . . . . . . . . . . . . . . . . . . . 511 9.5 Additional Information and Full Hypothesis Test Examples . . . . . . . . . . . . . . . . . . 514 9.6 Hypothesis Testing of a Single Mean and Single Proportion . . . . . . . . . . . . . . . . . . 530 Chapter10: Hypothesis Testing with Two Samples . . . . . . . . . . . . . . . . . . . . . . . . . 567 10.1 Two Population Means with Unknown Standard Deviations . . . . . . . . . . . . . . . . . 568 10.2 Two Population Means with Known Standard Deviations . . . . . . . . . . . . . . . . . . 576 10.3 Comparing Two Independent Population Proportions . . . . . . . . . . . . . . . . . . . . 579 10.4 Matched or Paired Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 10.5 Hypothesis Testing for Two Means and Two Proportions . . . . . . . . . . . . . . . . . . . 590 Chapter11: The Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 11.1 Facts About the Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 11.2 Goodness-of-Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 11.3 Test of Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 11.4 Test for Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 11.5 Comparison of the Chi-Square Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 11.6 Test of a Single Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 11.7 Lab 1: Chi-Square Goodness-of-Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 11.8 Lab 2: Chi-Square Test of Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 Chapter12: Linear Regression and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 12.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 12.2 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 12.3 The Regression Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 12.4 Testing the Significance of the Correlation Coefficient . . . . . . . . . . . . . . . . . . . . 691 12.5 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 12.6 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 12.7 Regression (Distance from School) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 12.8 Regression (Textbook Cost) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 12.9 Regression (Fuel Efficiency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 Chapter13: F Distribution and One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 13.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 13.2 The F Distribution and the F-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 13.3 Facts About the F Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 13.4 Test of Two Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 13.5 Lab: One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 AppendixA:Review Exercises (Ch 3-13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 AppendixB:Practice Tests (1-4) and Final Exams . . . . . . . . . . . . . . . . . . . . . . . . . . 813 AppendixC:Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 AppendixD:Group and Partner Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 AppendixE:Solution Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 AppendixF:Mathematical Phrases, Symbols, and Formulas . . . . . . . . . . . . . . . . . . . . 883 AppendixG:Notes for the TI-83, 83+, 84, 84+ Calculators . . . . . . . . . . . . . . . . . . . . . . 889 AppendixH:Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903 This OpenStax book is available for free at http://cnx.org/content/col11562/1.18 Preface 1 PREFACE Welcome to Introductory Statistics, an OpenStax resource. This textbook was written to increase student access to high- quality learning materials, maintaining highest standards of academic rigor at little to no cost. The foundation of this textbook is Collaborative Statistics, by Barbara Illowsky and Susan Dean. Additional topics, examples,andinnovationsinterminologyandpracticalapplicationshavebeenadded,allwithagoalofincreasingrelevance and accessibility for students. About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used byhundreds ofthousands ofstudents. OpenStax Tutor,ourlow-cost personalized learning tool, is being used in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliancewithothereducationalresourceorganizations,OpenStaxisbreakingdownthemostcommonbarrierstolearning and empowering students and instructors to succeed. About OpenStax's resources Customization IntroductoryStatisticsislicensedunderaCreativeCommonsAttribution4.0International(CCBY)license,whichmeans thatyoucandistribute,remix,andbuilduponthecontent,aslongasyouprovideattributiontoOpenStaxanditscontent contributors. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevanttotheneedsofyourcourse.Feelfreetoremixthecontentbyassigningyourstudentscertainchaptersandsections inyoursyllabus,intheorderthatyouprefer.Youcanevenprovideadirectlinkinyoursyllabustothesectionsintheweb view of your book. InstructorsalsohavetheoptionofcreatingacustomizedversionoftheirOpenStaxbook.Thecustomversioncanbemade availabletostudentsinlow-costprintordigitalformthroughtheircampusbookstore.VisityourbookpageonOpenStax.org for more information. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimesoccur.Sinceourbooksarewebbased,wecanmakeupdatesperiodicallywhendeemedpedagogicallynecessary. Ifyouhaveacorrectiontosuggest,submititthroughthelinkonyourbookpageonOpenStax.org.Subjectmatterexperts reviewallerratasuggestions.OpenStaxiscommittedtoremainingtransparentaboutallupdates,soyouwillalsofindalist of past errata changes on your book page on OpenStax.org. Format YoucanaccessthistextbookforfreeinwebvieworPDFthroughOpenStax.org,andinlow-costprintandiBookseditions. Coverage and scope Chapter 1 Sampling and Data Chapter 2 Descriptive Statistics Chapter 3 Probability Topics Chapter 4 Discrete Random Variables Chapter 5 Continuous Random Variables Chapter 6 The Normal Distribution Chapter 7 The Central Limit Theorem Chapter 8 Confidence Intervals Chapter 9 Hypothesis Testing with One Sample Chapter 10 Hypothesis Testing with Two Samples Chapter 11 The Chi-Square Distribution Chapter 12 Linear Regression and Correlation Chapter 13 F Distribution and One-Way ANOVA 2 Preface Alternate sequencing Introductory Statistics was conceived and written to fit a particular topical sequence, but it can be used flexibly to accommodate othercoursestructures. Onesuchpotential structure, whichfitsreasonably wellwiththetextbookcontent, isprovidedbelow.Pleaseconsider,however,thatthechapterswerenotwrittentobecompletelyindependent,andthatthe proposed alternate sequence should be carefully considered for student preparation and textual consistency. Chapter 1 Sampling and Data Chapter 2 Descriptive Statistics Chapter 12 Linear Regression and Correlation Chapter 3 Probability Topics Chapter 4 Discrete Random Variables Chapter 5 Continuous Random Variables Chapter 6 The Normal Distribution Chapter 7 The Central Limit Theorem Chapter 8 Confidence Intervals Chapter 9 Hypothesis Testing with One Sample Chapter 10 Hypothesis Testing with Two Samples Chapter 11 The Chi-Square Distribution Chapter 13 F Distribution and One-Way ANOVA Pedagogical foundation and features • Examples are placed strategically throughout the text to show students the step-by-step process of interpreting and solving statistical problems. To keep the text relevant for students, the examples are drawn from a broad spectrum ofpracticaltopics,includingexamplesaboutcollegelifeandlearning,healthandmedicine,retailandbusiness,and sports and entertainment. • TryItpracticeproblemsimmediatelyfollowmanyexamplesandgivestudentstheopportunitytopracticeastheyread the text.They are usually based on practical and familiar topics, like the Examples themselves. • Collaborative Exercisesprovide an in-class scenario for students to work together to explore presented concepts. • Using the TI-83, 83+, 84, 84+ Calculator shows students step-by-step instructions to input problems into their calculator. • The Technology Iconindicates where the use of a TI calculator or computer software is recommended. • Practice,Homework,andBringingItTogetherproblemsgivethestudentsproblemsatvariousdegreesofdifficulty while also including real-world scenarios to engage students. Statistics labs TheseinnovativeactivitiesweredevelopedbyBarbaraIllowskyandSusanDeaninordertoofferstudentstheexperienceof designing,implementing,andinterpretingstatisticalanalyses.Theyaredrawnfromactualexperimentsanddata-gathering processesandofferauniquehands-onandcollaborativeexperience.Thelabsprovideafoundationforfurtherlearningand classroom interaction that will produce a meaningful application of statistics. StatisticsLabsappearattheendofeachchapterandbeginwithstudentlearningoutcomes,generalestimatesfortimeon task,andanyglobalimplementation notes.Studentsarethenprovidedwithstep-by-stepguidance,includingsampledata tablesandcalculationprompts.Thedetailedassistancewillhelpthestudentssuccessfullyapplytheconceptsinthetextand lay the groundwork for future collaborative or individual work. 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