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Easily move from one to another and your students Online Exercises were carefully selected and developed to provide a seamless transition from textbook to technology For consistency, the guided solutions match the style and voice of the original text as though the author is guiding the students through the problems www.mheducation.com ALLAN G BLUMAN PROFESSOR EMERITUS COMMUNITY COLLEGE OF ALLEGHENY COUNTY ELEMENTARY STATISTICS: A STEP BY STEP APPROACH, TENTH EDITION Published by McGraw-Hill Education, Penn Plaza, New York, NY 10121 Copyright © 2018 by McGraw-Hill Education All rights reserved Printed in the United States of America Previous editions © 2014, 2012, and 2009 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper LWI 21 20 19 18 17 ISBN 978-1-259-75533-0 MHID 1-259-75533-9 ISBN 978-1-260-04200-9 (Annotated Instructor’s Edition) MHID 1-260-04200-6 Chief Product Officer, SVP Products & Markets: G Scott Virkler Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Design & Delivery: Betsy Whalen Managing Director: Ryan Blankenship Brand Manager: Adam Rooke Freelance Product Developer: Christina Sanders Director, Product Development: Rose Koos Marketing Director: Sally Yagan Digital Product Analysts: Ruth Czarnecki-Lichstein and Adam Fischer Director, Digital Content: Cynthia Northrup Director, Content Design & Delivery: Linda Avenarius Program Manager: Lora Neyens Content Project Managers: Jane Mohr, Emily Windelborn, and Sandra Schnee Buyer: Sandy Ludovissy Design: Matt Backhous Content Licensing Specialists: Lorraine Buczek and Melissa Homer Cover Image: © Kim Doo-Ho/VisionsStyler Press/Getty Images RF Compositor: MPS Limited Printer: LSC Communications All credits appearing on page or at the end of the book are considered to be an extension of the copyright page Library of Congress Cataloging-in-Publication Data Bluman, Allan G Elementary statistics : a step by step approach / Allan G Bluman, professor emeritus, Community College of Allegheny Dounty Tenth edition | New York, NY : McGraw-Hill Education, [2018] | Includes index LCCN 2016028437 | ISBN 9781259755330 (alk paper) LCSH: Statistics—Textbooks | Mathematical statistics—Textbooks LCC QA276.12 B59 2018 | DDC 519.5—dc23 LC record available at https://lccn.loc.gov/2016028437 The Internet addresses listed in the text were accurate at the time of publication The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites mheducation.com/highered ABOUT THE AUTHOR Allan G Bluman Allan G Bluman is a professor emeritus at the Community College of Allegheny County, South Campus, near Pittsburgh, Pennsylvania He has taught mathematics and statistics for over 35 years He received an Apple for the Teacher award in recognition of his bringing excellence to the learning environment at South Campus He has also taught statistics for Penn State University at the Greater Allegheny (McKeesport) Campus and at the Monroeville Center He received his master’s and doctor’s degrees from the University of Pittsburgh Courtesy Allan G Bluman He is also author of Elementary Statistics: A Brief Version and coauthor of Math in Our World In addition, he is the author of four mathematics books in the McGraw-Hill DeMystified Series They are Pre-Algebra, Math Word Problems, Business Math, and Probability He is married and has two sons, a granddaughter, and a grandson Dedication: To Betty Bluman, Earl McPeek, and Dr G Bradley Seager, Jr iii This page intentionally left blank CONTENTS Preface ix C H A P T E R The Nature of Probability and Statistics Introduction 1–1 Descriptive and Inferential Statistics 1–2 Variables and Types of Data 1–3 Data Collection and Sampling Techniques 11 Random Sampling 12 Systematic Sampling 12 Stratified Sampling 13 Cluster Sampling 14 Other Sampling Methods 14 1–4 Experimental Design 18 Observational and Experimental Studies 18 Uses and Misuses of Statistics 21 1–5 Computers and Calculators 26 Summary 33 C H A P T E R Frequency Distributions and Graphs 41 Introduction 42 2–1 Organizing Data 42 Categorical Frequency Distributions 43 Grouped Frequency Distributions 44 2–2 Histograms, Frequency Polygons, and Ogives 57 The Histogram 57 The Frequency Polygon 58 The Ogive 59 Relative Frequency Graphs 61 Distribution Shapes 63 2–3 Other Types of Graphs 74 Bar Graphs 75 Pareto Charts 77 The Time Series Graph 78 The Pie Graph 80 Dotplots 83 Stem and Leaf Plots 83 Misleading Graphs 86 Summary 100 C H A P T E R Data Description 109 Introduction 110 3–1 Measures of Central Tendency 111 The Mean 111 The Median 114 The Mode 116 The Midrange 118 The Weighted Mean 119 Distribution Shapes 121 3–2 Measures of Variation 127 Range 129 Population Variance and Standard Deviation 129 Sample Variance and Standard Deviation 132 Variance and Standard Deviation for Grouped Data 135 Coefficient of Variation 137 Range Rule of Thumb 138 Chebyshev’s Theorem 139 The Empirical (Normal) Rule 141 Linear Transformation of Data 142 3–3 Measures of Position 148 Standard Scores 148 All examples and exercises in this textbook (unless cited) are hypothetical and are presented to enable students to achieve a basic understanding of the statistical concepts explained These examples and exercises should not be used in lieu of medical, psychological, or other professional advice Neither the author nor the publisher shall be held responsible for any misuse of the information presented in this textbook v vi Contents Percentiles 149 Quartiles and Deciles 155 Outliers 157 3–4 Exploratory Data Analysis 168 The Five-Number Summary and Boxplots 168 Summary 177 C H A P T E R Probability and Counting Rules 185 Introduction 186 4–1 Sample Spaces and Probability 186 Basic Concepts 186 Classical Probability 189 Complementary Events 192 Empirical Probability 194 Law of Large Numbers 196 Subjective Probability 196 Probability and Risk Taking 196 4–2 The Addition Rules for Probability 201 4–3 The Multiplication Rules and Conditional Probability 213 4–4 The Multiplication Rules 213 Conditional Probability 217 Probabilities for “At Least” 220 Counting Rules 226 The Fundamental Counting Rule 227 Factorial Notation 229 Permutations 229 Combinations 232 4–5 Probability and Counting Rules 242 Summary 246 C H A P T E R Discrete Probability Distributions 257 Introduction 258 5–1 Probability Distributions 258 5–2 Mean, Variance, Standard Deviation, and Expectation 265 Mean 265 Variance and Standard Deviation 267 Expectation 269 5–3 The Binomial Distribution 275 5–4 Other Types of Distributions 289 The Multinomial Distribution 289 The Poisson Distribution 291 The Hypergeometric Distribution 293 The Geometric Distribution 295 Summary 303 C H A P T E R The Normal Distribution 311 Introduction 312 6–1 Normal Distributions 312 The Standard Normal Distribution 315 Finding Areas Under the Standard Normal Distribution Curve 316 A Normal Distribution Curve as a Probability Distribution Curve 318 6–2 Applications of the Normal Distribution 328 6–3 Finding Data Values Given Specific Probabilities 332 Determining Normality 334 The Central Limit Theorem 344 Distribution of Sample Means 344 Finite Population Correction Factor (Optional) 350 6–4 The Normal Approximation to the Binomial Distribution 354 Summary 361 C H A P T E R Confidence Intervals and Sample Size 369 Introduction 370 7–1 Confidence Intervals for the Mean When σ Is Known 370 7–2 Confidence Intervals 371 Sample Size 377 Confidence Intervals for the Mean When σ Is Unknown 383 Contents 7–3 Confidence Intervals and Sample Size for Proportions 390 Confidence Intervals 391 Sample Size for Proportions 393 vii 9–5 Testing the Difference Between Two Variances 528 Summary 539 10 7–4 C H A P T E R C H A P T E R Correlation and Regression 547 Confidence Intervals for Variances and Standard Deviations 398 Summary 406 Hypothesis Testing 413 Introduction 548 10–1 Scatter Plots and Correlation 548 Introduction 414 8–1 Steps in Hypothesis Testing—Traditional Method 414 8–2 z Test for a Mean 426 P-Value Method for Hypothesis Testing 430 8–3 t Test for a Mean 442 8–4 z Test for a Proportion 453 8–5 𝛘2 Test for a Variance or Standard Deviation 461 8–6 Additional Topics Regarding Hypothesis Testing 474 Confidence Intervals and Hypothesis Testing 474 Type II Error and the Power of a Test 476 Summary 479 C H A P T E R Testing the Difference Between Two Means, Two Proportions, and Two Variances 487 Introduction 488 9–1 Testing the Difference Between Two Means: Using the z Test 488 9–2 Testing the Difference Between Two Means of Independent Samples: Using the t Test 499 9–3 Testing the Difference Between Two Means: Dependent Samples 507 9–4 Testing the Difference Between Proportions 519 Correlation 552 10–2 Regression 563 Line of Best Fit 564 Determination of the Regression Line Equation 565 10–3 Coefficient of Determination and Standard Error of the Estimate 580 Types of Variation for the Regression Model 580 Residual Plots 582 Coefficient of Determination 583 Standard Error of the Estimate 584 Prediction Interval 587 10–4 Multiple Regression (Optional) 590 The Multiple Regression Equation 591 Testing the Significance of R 593 Adjusted R2 594 Summary 599 C H A P T E R 11 Other Chi-Square Tests 607 Introduction 608 11–1 Test for Goodness of Fit 608 Test of Normality (Optional) 614 11–2 Tests Using Contingency Tables 622 Test for Independence 622 Test for Homogeneity of Proportions 628 Summary 638 viii Contents C H A P T E R 12 Analysis of Variance 645 Introduction 646 12–1 One-Way Analysis of Variance 646 12–2 The Scheffé Test and the Tukey Test 658 Scheffé Test 658 Tukey Test 659 12–3 Two-Way Analysis of Variance 662 Summary 676 C H A P T E R 13 Nonparametric Statistics 685 C H A P T E R 14 Sampling and Simulation 737 Introduction 738 14–1 Common Sampling Techniques 738 Random Sampling 739 Systematic Sampling 742 Stratified Sampling 744 Cluster Sampling 746 Other Types of Sampling Techniques 746 14–2 Surveys and Questionnaire Design 753 14–3 Simulation Techniques and the Monte Carlo Method 756 The Monte Carlo Method 756 Summary 762 APPENDICES Introduction 686 13–1 Advantages and Disadvantages of Nonparametric Methods 686 Advantages 686 Disadvantages 686 Ranking 687 A Tables 769 B Data Bank 794 C Glossary 801 D Selected Answers SA–1 13–2 The Sign Test 689 Single-Sample Sign Test 689 Paired-Sample Sign Test 691 13–3 The Wilcoxon Rank Sum Test 698 13–4 The Wilcoxon Signed-Rank Test 703 13–5 The Kruskal-Wallis Test 708 13–6 The Spearman Rank Correlation Coefficient and the Runs Test 715 Rank Correlation Coefficient 715 The Runs Test 718 Summary 729 Index I–1 ADDITIONAL TOPICS ONLINE (www.mhhe.com/bluman) Algebra Review Writing the Research Report Bayes’ Theorem Alternate Approach to the Standard Normal Distribution Bibliography Appendix D Selected Answers Biased 10 Cluster 11–14. Answers will vary 15 Use two-digit random numbers: 01 through 45 means the player wins Any other two-digit random number means the player loses 16 Use two-digit random numbers: 01 through 05 means a cancellation Any other two-digit random number means the person shows up 17 The random numbers 01 through 10 represent the 10 cards in hearts The random numbers 11 through 20 represent the 10 cards in diamonds The random numbers 21 through 30 represent the 10 spades, and 31 through 40 represent the 10 clubs Any number over 40 is ignored 18 Use two-digit random numbers to represent the spots on the face of the dice Ignore any two-digit random numbers SA–44 with 7, 8, 9, or For cards, use two-digit random numbers between 01 and 13 19 Use two-digit random numbers The first digit represents the first player, and the second digit represents the second player If both numbers are odd or even, player wins If a digit is odd and the other digit is even, player 2 wins 20–24. Answers will vary 25 Here regularly is vague 26 Bad weather means different things to different people 27 What is meant by readable? 28 Smoking a lot means different things to different people 29 Some respondents might not know much about herbal medicine 30 Almost everybody would answer “No” to this question INDEX A Addition rules, 201–206 Adjusted R2, 594 Alpha, 419 Alternative hypotheses, 414 Analysis of variance (ANOVA), 646–653 assumptions, 648 between-group variance, 647 degrees of freedom, 648 F-test, 647 hypotheses, 646, 665 one-way, 646–653 summary table, 648 two-way, 662–670 within-group variance, 647 Assumption for the use of the chi-square test, 464, 609 Assumptions, 370 Assumptions for valid predictions in regression, 568 Averages, 111–121 properties and uses, 120–121 B Bar graph, 75–77 Bell curve, 312 Beta, 419 Between-group variance, 647 Biased sample, 3, 738 Bimodal, 64, 116 Binomial distribution, 276–282 characteristics, 275 mean for, 280 normal approximation, 354–359 notation, 276 standard deviation, 280–281 variance, 280–281 Binomial experiment, 275 Binomial probability formula, 277 Blinding, 20 Blocks, 20 Boundaries, Boundaries, class, 45 Boundary, Boxplot, 168–171 C Categorical frequency distribution, 43–44 Census, Central limit theorem, 344–354 Chebyshev’s theorem, 139–141 Chi-square assumptions, 464, 609 contingency table, 622 degrees of freedom, 400 distribution, 398–401, 608 goodness-of-fit test, 608–617 independence test, 622–628 use in H-test, 709 variance test, 461–469 Yates correction for, 630 Class, 42 boundaries, 7, 45 limits, 45 midpoint, 45 width, 45 Classical probability, 189–193 Cluster sample, 14, 746 Coefficient of determination, 583–584 Coefficient of nondetermination, 584 Coefficient of variation, 137–138 Combination, 232–234 Combination rule, 233 Complementary events, 192–193 Complement of an event, 192 Completely randomized designs, 20 Compound bar graph, 76–77 Compound event, 189 Conditional probability, 215, 217–220 Confidence interval, 371 hypothesis testing, 474–476 mean, 372–377, 383–386 means, differences of, 493, 501, 514 median, 696 proportion, 391–393 proportions, differences, 523–524 variances and standard deviations, 398–403 Confidence level, 371 Confounding variable, 19 Consistent estimator, 371 Contingency coefficient, 635 Contingency table, 622 Continuous variable, 6, 258, 312 Control group, 19 Convenience sample, 14, 746 Correction for continuity, 354 Correlation, 552–560 Correlation coefficient, 552 multiple, 592–593 Pearson’s product moment, 552 population, 552 Spearman’s rank, 715–718 Critical region, 420 Critical value, 420, 422–424 Cross-sectional study, 18 Cumulative frequency, 59 Cumulative frequency distribution, 48–49 Cumulative frequency graph, 59 Cumulative relative frequency, 62 D Data, Data array, 114 Data set, Data transformation, 142 Data value (datum), Deciles, 157 Degrees of freedom, 383, 442 Dependent events, 215 Dependent samples, 488, 507 Dependent variable, 19, 488, 507, 508, 548, 663 Descriptive statistics, Difference between two means, 488–493, 499–502, 507–513 assumptions for the test to determine, 489, 500, 509 proportions, 519–523 Discrete probability distributions, 259 Discrete variable, 6, 258 Disjoint events, 202 Disordinal interaction, 669 Distribution-free statistics (nonparametric), 686 Distributions bell-shaped, 63, 312 bimodal, 64, 116 binomial, 276–282 chi-square, 399–401 F, 529 frequency, 42 geometric, 295–297 hypergeometric, 293–295 multinomial, 290–291 negatively skewed, 64, 122 normal, 312–321 Poisson, 291–293 positively skewed, 63–64, 121, 315 probability, 258, 263 sampling, 344 standard normal, 315–318 symmetrical, 63, 121, 314 Dotplot, 83 Double blinding, 20 Double sampling, 746 E Empirical probability, 194–196 Empirical rule, 141, 314 Equally likely events, 189 Estimation, 370 Estimator, properties of a good, 371 Event, 188 Event, simple, 189 I–1 Index Events complementary, 192–193 compound, 189 dependent, 215 disjoint, 202 equally likely, 189 independent, 213 mutually exclusive, 202 Expectation, 269–272 Expected frequency, 608 Expected value, 269 Experimental study, 18 Explained variation, 19, 580 Explanatory variable, 19, 548 Exploratory data analysis (EDA), 168–171 Extrapolation, 569 F Factorial notation, 229 Factors, 662 F-distribution, characteristics of, 529, 646 Finite population correction factor, 350–351 Five-number summary, 168 Frequency, 42 Frequency distribution, 42 categorical, 43–44 grouped, 44–48 reasons for, 50–51 rules for constructing, 45–46 ungrouped, 49–50 Frequency polygon, 58–59 F-test, 528–531, 650 comparing three or more means, 648 comparing two variances, 531–534 notes for the use of, 531 Fundamental counting rule, 226–229 definitions, 414 level of significance, 419 noncritical region, 420 null, 414 one-tailed test, 420 P-value method, 430–434 research, 415 statistical, 414 statistical test, 417 test value, 417, 426 traditional method, steps in, 424 two-tailed test, 421, 422 types of errors, 418–419 I Independence test (chi-square), 622–628 Independent events, 213 Independent samples, 488, 499 Independent variables, 19, 548, 663 Inferential statistics, Influential observation or point, 569 Interaction effect, 664 Intercept ( y), 565–568 Interquartile range (IQR), 156 Interval estimate, 371 Interval level of measurement, K Kruskal-Wallis test, 708–711 L Gallup poll, 738 Gaussian distribution, 312 Geometric distribution, 295–297 Geometric experiment, 296 Geometric mean, 126 Goodness-of-fit test, 608–614 Grand mean, 647 Grouped frequency distribution, 44–48 Law of large numbers, 196 Left-tailed test, 420–422 Level of significance, 419 Levels of measurement, interval, nominal, ordinal, ratio, Limits, class, 44 Line of best fit, 564 Longitudinal study, 18 Lower class boundary, 45 Lower class limit, 44 Lurking variable, 19, 560 H M Harmonic mean, 126 Hawthorne effect, 19 Hinges, 171 Histogram, 57–58 Homogeneity or proportions, 628–630 Homoscedasticity assumption, 583 Hypergeometric distribution, 293–295 Hypergeometric experiment, 294 Hypothesis, 4, 414 Hypothesis testing, 4, 414–425 alternative, 414 common phrases, 416 critical region, 420 critical value, 420 Main effects, 664 Marginal change, 569 Margin of error, 372 Matched pair design, 20 Mean, 111–114 binomial variable, 281–282 definition, 112 population, 112 probability distribution, 265–267 sample, 112 Mean deviation, 147 Mean square, 648 Measurement, levels of, Measurement scales, G I–2 Measures of average, uses of, 120–121 Measures of dispersion, 127–137 Measures of position, 148–157 Measures of variation, 127–137 Measures of variation and standard deviation, uses of, 137 Median, 114–116 confidence interval for, 696 defined, 114 for grouped data, 126–127 Midquartile, 161 Midrange, 118–119 Misleading graphs, 23, 86–89 Modal class, 117 Mode, 116–118 Modified boxplot, 171, 173 Monte Carlo method, 756–760 Multimodal, 116 Multinomial distribution, 290–291 Multinomial experiment, 290 Multiple correlation coefficient, 592–593 Multiple regression, 590–595 Multiplication rules probability, 213–217 Multistage sampling, 746 Mutually exclusive events, 202 N Negative linear relationship, 549, 552 Negatively skewed distribution, 122, 315 Nielson television ratings, 738 Nominal level of measurement, Noncritical region, 420 Nonparametric statistics, 686–729 advantages, 686 disadvantages, 686–687 Nonrejection region, 420 Nonresistant statistic, 157 Nonresponse bias, 738 Nonsampling error, 16 Normal approximation to binomial distribution, 354–359 Normal distribution, 312–321 applications of, 328–334 approximation to the binomial distribution, 354–359 areas under, 314–315 formula for, 313 probability distribution as a, 318–320 properties of, 314 standard, 315–318 Normally distributed variables, 312–315 Normal quantile plot, 337, 342 Notation for the binomial distribution, 276 Null hypothesis, 414 O Observational study, 18 Observed frequency, 608 Odds, 201 Ogive, 59–61 One-tailed test, 420 left, 420 right, 420 Index One-way analysis of variance, 646 Open-ended distribution, 46 Ordinal interaction, 669 Ordinal level of measurement, Outcome, 186 Outcome variable, 19 Outliers, 64, 118, 157–158, 335 P Paired-sample sign test, 691–693 Parameter, 111 Parametric tests, 686 Pareto chart, 77–78 Pearson coefficient of skewness, 147, 335 Pearson product moment correlation coefficient, 552 Percentiles, 149–155 Permutation, 229–231 Permutation rule 1, 230 Permutation rule 2, 231 Pie graph, 80–83 Placebo effect, 20 Point estimate, 370 Poisson distribution, 291–293 Poisson experiment, 291 Pooled estimate of variance, 502 Population, 3, 738 Population correlation coefficient, 552 Positive linear relationship, 549, 552 Positively skewed distribution, 121, 315 Power of a test, 476 Practical significance, 434 Prediction interval, 584, 587–589 Probability, 4, 186 addition rules, 201–206 at least, 220–221 binomial, 275–280 classical, 189–193 complimentary rules, 193 conditional, 215, 217–220 counting rules, 242–243 distribution, 258–263 empirical, 194–196 experiment, 186 multiplication rules, 213–217 subjective, 196 Properties of the distribution of sample means, 344 Proportion, 61, 390–394 P-value, 431 for χ2 test, 466–468 for F test, 533 method for hypothesis testing, 452–456 for t test, 446 Q Quadratic mean, 126 Qualitative variables, Quantile plot, 337, 342–343 Quantitative variables, Quartiles, 155–157 Quasi-experimental study, 19 Questionnaire design, 753–754 R Random numbers, 12 Random samples, 12, 740 Random sampling, 12, 739–742 Random variable, 3, 258 Range, 47, 129 Range rule of thumb, 139 Rank correlation, Spearman’s, 715–718 Ranking, 687–688 Ratio level of measurement, Raw data, 42 Regression, 563–569 assumptions for valid prediction, 568 multiple, 590–595 Regression line, 563 equation, 565 intercept, 565 line of best fit, 564 prediction, 568 slope, 565 Rejection region, 420 Relationships, 4, 548 Relative frequency graphs, 61–63 Relatively efficient estimator, 371 Replication, 20 Requirements for a probability distribution, 261 Research hypothesis, 415 Residual, 564 Residual plot, 582–583 Resistant statistic, 157 Response bias, 738 Response variable, 550 Retrospective study, 18 Right-tailed test, 420, 422 Robust statistical technique, 373 Run, 718 Runs test, 718–723 S Sample, 3, 738 biased, 738 cluster, 14, 746 convenience, 14 random, 12, 738 size for estimating means, 377–378 size for estimating proportions, 393–395 stratified, 14, 744–746 systematic, 12–13 unbiased, 738 volunteer, 14 Sample space, 186–187 Sampling, 3, 12–14, 738–747 distribution of sample means, 344 double, 746 error, 14, 16, 344 multistage, 746 random, 12, 738 sequence, 746 Scatter plot, 548–552 Scheffé test, 658–661 Selection bias, 738 Sequence sampling, 746 Shortcut formula for variance and standard deviation, 134–135 Significance, level of, 419 Sign test, 689 test value, 689–691 Simple event, 189 Simulation technique, 738, 756–760 Single sample sign test, 689–691 Skewness, 63–64 Slope, 565 Spearman rank correlation coefficient, 715–718 Standard deviation, 130–138 binomial distribution, 280–281 definition, 130, 133 formula, 130, 133 population, 130 probability distribution, 267–269 sample, 133 uses of, 138 Standard error of difference between means, 490 Standard error of difference between proportions, 520 Standard error of the estimate, 584–587 Standard error of the mean, 346 Standard normal distribution, 315–318 Standard score, 148–149 Statistic, 111 Statistical hypothesis, 414 Statistical test, 417 Statistics, descriptive, inferential, misuses of, 21–23 Stem and leaf plot, 83–86 Stratified sample, 13, 744–746 Student’s t distribution, 383 Subjective probability, 196 Sum of squares, 648 Surveys, 11, 753–754 mail, 11 personal interviews, 11 telephone, 11 Symmetric distribution, 63, 121, 314 Systematic sampling, 12–13, 742–744 T t-distribution, characteristics of, 383, 442 Test of normality, 334–337, 343, 614–616 Test value, 417, 426 Time series graph, 78–79 Total variation, 580 Treatment groups, 19, 663 Tree diagram, 188, 217, 227, 228 t-test, 442 coefficient for correlation, 552 for difference of means, 499–502, 507–513 for mean, 442–448 Tukey test, 658, 659–661 Two-tailed test, 421, 422 Two-way analysis of variance, 662–670 Type I error, 418, 476–478 Type II error, 418, 476–478 I–3 Index U Unbiased estimate of population variance, 133 Unbiased estimator, 371 Unbiased sample, 738 Unexplained variation, 580 Ungrouped frequency distribution, 49–50 Uniform distribution, 63, 321 Unimodal, 64, 116 Upper class boundary, 45 Upper class limit, 44–45 V Variable, 3, 258, 548 confounding, 19 continuous, 6, 258, 312 dependent, 19, 488, 507, 508, 548 discrete, 6, 258 explanatory, 19, 548 independent, 19, 548 I–4 outcome, 19 qualitative, quantitative, random, 3, 258 response, 548 Variance, 130–138 binomial distribution, 280–281 definition of, 130, 133 formula, 130, 133 population, 130 probability distribution, 265–267 sample, 133 shortcut formula, 134 unbiased estimate, 133 uses of, 138 Variances equal, 528–529 unequal, 528–529 Venn diagram, 193, 203, 204, 218 Volunteer bias, 738 Volunteer sample, 14 W Weighted estimate of p, 520 Weighted mean, 119–120 Wilcoxon rank sum test, 698–700 Wilcoxon signed-rank test, 703–706 Within-group variance, 647 Y Yates correction for continuity, 630 y-intercept, 565–568 Z z-score, 148–149, 316 z-test, 427 z-test for means, 426–430, 488–493 z-test for proportions, 452–456, 519–523 z-values (scores), 316 Glossary of Symbols a α b β C cf C n r y intercept of a line Probability of a type I error Slope of a line Probability of a type II error Column frequency Cumulative frequency Number of combinations of n objects taking r objects at a time C.V Critical value CVar Coefficient of variation D Difference; decile D Mean of the differences d.f Degrees of freedom d.f.N Degrees of freedom, numerator d.f.D Degrees of freedom, denominator E Event; expected frequency; maximum error of estimate E Complement of an event e Euler’s constant ≈ 2.7183 E(X) Expected value f Frequency F F test value; failure F′ Critical value for the Scheffé test MD Median MR Midrange MSB Mean square between groups MSW Mean square within groups (error) n Sample size N Population size n(E) Number of ways E can occur n(S) Number of outcomes in the sample space O Observed frequency P Percentile; probability p Probability; population proportion P ˆ Sample proportion p Weighted estimate of p P(B A ⃒ ) Conditional probability P(E) Probability of an event E P(E ) Probability of the complement of E P Number of permutations of n objects taking n r r objects at a time π Pi ≈ 3.14 Q Quartile q 1 − p; test value for Tukey test ˆ 1 − p ˆ q q 1 − p R Range; rank sum FS Scheffé test value GM Geometric mean H Kruskal-Wallis test value H0 Null hypothesis H1 Alternative hypothesis HM Harmonic mean k Number of samples λ Number of occurrences for the Poisson distribution sD Standard deviation of the differences sest Standard error of estimate SSB Sum of squares between groups SSW Sum of squares within groups sB Between-group variance sW Within-group variance t t test value tα∕2 Two-tailed t critical value μ Population mean μD Mean of the population differences _ μ X Mean of the sample means w Class width; weight r Sample correlation coefficient R Multiple correlation coefficient r Coefficient of determination 𝜌 Population correlation coefficient rS Spearman rank correlation coefficient S Sample space; success s Sample standard deviation s2 Sample variance σ Population standard deviation σ Population variance _ σ X Standard error of the mean Σ Summation notation ws Smaller sum of signed ranks, Wilcoxon signed‑rank test X Data value; number of successes for a binomial distribution X Sample mean x Independent variable in regression X GM Grand mean Xm Midpoint of a class � 2 Chi-square y Dependent variable in regression y′ Predicted y value z z test value or z score zα∕2 Two-tailed critical z value ! Factorial www.freebookslides.com Important Formulas Chapter 5 Discrete Probability Distributions Mean for a probability distribution: 𝜇 = Σ [X · P(X)] Variance and standard deviation for a probability distribution: Chapter 3 Data Description = _ ΣX Mean for individual data: X n Σf · Xm Mean for grouped data: X = _ n 𝜎2 = Σ [X 2 · P(X)] − 𝜇2 Σ(X − X )2 n(Σ X 2) − (Σ X )2 s = _ or s = n−1 n(n − 1) _ √ √ 𝜎 = √ Σ [X 2 · P(X)] − 𝜇2 _ Standard deviation for a sample: √ (Shortcut formula) Standard deviation for grouped data: _ n(Σf · X2m ) − (Σf · X m ) _ s = n(n − 1) range Range rule of thumb: s ≈ _ Chapter 4 Probability and Counting Rules Addition rule (mutually exclusive events): P(A or B ) = P(A) + P(B) Addition rule (events not mutually exclusive): P(A or B) = P(A) + P(B ) − P(A and B) Expectation: E (X ) = Σ [X · P(X)] n! Binomial probability: P(X ) = _ · pX · qn − X (n − X )!X! Mean for binomial distribution: 𝜇 = n · p Variance and standard deviation for the binomial distribution: _ 𝜎2 = n · p · q 𝜎 = √n · p · q Multinomial probability: X X X X n! · p 1 · p2 2 · p3 3 · · · pk k P(X) = _ X1! X2! X3! · · · Xk! −𝜆 X Poisson probability: P(X; 𝜆) = e 𝜆 where X! X = 0, 1, 2, C · C −X a X b n Hypergeometric probability: P(X ) = a+bCn Geometric probability: P(n) = p(1 − p)n − where n is the number of the trial in which the first success occurs and p is the probability of a success Multiplication rule (independent events): P(A and B) = P(A) · P(B) P(A and B) = P(A) · P(B | A) Multiplication rule (dependent events): P(A and B) Conditional probability: P(B | A) = P(A) Complementary events: P(E ) = − P(E ) Fundamental counting rule: Total number of outcomes of a sequence when each event has a different number of possibilities: k1 · k2 · k3 · · · kn Permutation rule: Number of permutations of n objects taking r at a time is nPr = _ n! (n − r)! Permutation rule of n objects with r1 objects identical, r2 objects identical, etc n! r1! r2! · · · rp! Combination rule: Number of combinations of r objects n! selected from n objects is nCr = _ (n − r)!r! blu55339_Important_Formulas.indd Chapter 6 The Normal Distribution X−μ X Standard score: Population: z = _ 𝜎 or Sample: z = X − s Mean of sample means: 𝜇_X = 𝜇 Standard error of the mean: 𝜎_X = 𝜎 √ n − μ X Central limit theorem formula: z = 𝜎∕√n Chapter 7 Confidence Intervals and Sample Size z confidence interval for means: σ < 𝜇 < σ X − zα⧸2 + zα⧸2 X √ n √n ( ) ( ) t confidence interval for means: s s − tα⧸2 X < 𝜇 < X + tα⧸2 √ √ n n zα⧸2 · 𝜎 Sample size for means: n = where E is the margin E of error Confidence interval for a proportion: ( ) ( ( ) ) p ˆ ˆq pˆ − (zα⧸2) n < p