STATISTICS WITH JMP: HYPOTHESIS TESTS, ANOVA AND REGRESSION PETER GOOS AND DAVID MEINTRUP STATISTICS WITH JMP: HYPOTHESIS TESTS, ANOVA AND REGRESSION STATISTICS WITH JMP HYPOTHESIS TESTS, ANOVA AND REGRESSION Peter Goos University of Leuven and University of Antwerp, Belgium David Meintrup University of Applied Sciences Ingolstadt, Germany This edition first published 2016 © 2016 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the authors to be identified as the authors of this work has been asserted in accordance withthe Copyright, Designs and Patents Act 1988. All rights reserved. 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Identifiers: LCCN 2015039990 (print) | LCCN 2015047679 (ebook) | ISBN 9781119097150 (cloth)| ISBN 9781119097044 (Adobe PDF) | ISBN 9781119097167 (ePub) Subjects: LCSH: Probabilities–Data processing. | Mathematical statistics–Data processing. | Regression analysis. | JMP (Computer file) Classification: LCC QA273.19.E4 G68 2016 (print) | LCC QA273.19.E4 (ebook)| DDC 519.50285/53–dc23 LC record available at http://lccn.loc.gov/2015039990 A catalogue record for this book is available from the British Library. Set in 10/12pt Times by Aptara Inc., New Delhi, India 1 2016 To Marijke, Bas, Loes, and Mien To Beatrice´ and Werner Contents Preface xv Acknowledgments xix Part One ESTIMATORS AND TESTS 1 1 Estimating Population Parameters 3 1.1 Introduction: Estimators Versus Estimates 3 1.2 Estimating a Mean Value 4 1.2.1 The Mean of a Normally Distributed Population 4 1.2.2 The Mean of an Exponentially Distributed Population 6 1.3 Criteria for Estimators 6 1.3.1 Unbiased Estimators 8 1.3.2 The Efficiency of an Estimator 9 1.4 Methods for the Calculation of Estimators 9 1.5 The Sample Mean 10 1.5.1 The Expected Value and the Variance 10 1.5.2 The Probability Density of the Sample Mean for a Normally Distributed Population 12 1.5.3 The Probability Density of the Sample Mean for a Nonnormally Distributed Population 12 1.5.4 An Illustration of the Central Limit Theorem 14 1.6 The Sample Proportion 18 1.7 The Sample Variance 21 1.7.1 The Expected Value 22 1.7.2 The 2-Distribution 24 1.7.3 The Relation Between the Standard Normal and the 2-Distribution 25 1.7.4 The Probability Density of the Sample Variance 28 1.8 The Sample Standard Deviation 30 1.9 Applications 33 2 Interval Estimators 35 2.1 Point and Interval Estimators 35 viii Contents 2.2 Confidence Intervals for a Population Mean with Known Variance 36 2.2.1 The Percentiles of the Standard Normal Density 36 2.2.2 Computing a Confidence Interval 38 2.2.3 The Width of a Confidence Interval 39 2.2.4 The Margin of Error 41 2.3 Confidence Intervals for a Population Mean with Unknown Variance 41 2.3.1 The Student t-Distribution 42 2.3.2 The Application of the t-Distribution to Construct Confidence Intervals 45 2.4 Confidence Intervals for a Population Proportion 47 2.4.1 A First Interval Estimator Based on the Normal Distribution 47 2.4.2 A Second Interval Estimator Based on the Normal Distribution 49 2.4.3 An Interval Estimator Based on the Binomial Distribution 50 2.5 Confidence Intervals for a Population Variance 52 2.6 More Confidence Intervals in JMP 56 2.7 Determining the Sample Size 58 2.7.1 The Population Mean 60 2.7.2 The Population Proportion 63 3 Hypothesis Tests 67 3.1 Key Concepts 67 3.2 Testing Hypotheses About a Population Mean 71 3.2.1 The Right-Tailed Test 71 3.2.2 The Left-Tailed Test 77 3.2.3 The Two-Tailed Test 78 3.3 The Probability of a Type II Error and the Power 85 3.4 Determination of the Sample Size 88 3.5 JMP 90 3.6 Some Important Notes Concerning Hypothesis Testing 93 3.6.1 Fixing the Significance Level 94 3.6.2 A Note on the “Acceptance” of the Null Hypothesis 95 3.6.3 Statistical and Practical Significance 95 Part Two ONE POPULATION 97 4 Hypothesis Tests for a Population Mean, Proportion, or Variance 99 4.1 Hypothesis Tests for One Population Mean 99 4.1.1 The Right-Tailed Test 100 4.1.2 The Left-Tailed Test 102 4.1.3 The Two-Tailed Test 102 4.1.4 Nonnormal Data 107 4.1.5 The Use of JMP 108 4.2 Hypothesis Tests for a Population Proportion 110 4.2.1 Tests Based on the Normal Distribution 111 4.2.2 Tests Based on the Binomial Distribution 117 4.2.3 Testing Proportions in JMP 119 Contents ix 4.3 Hypothesis Tests for a Population Variance 122 4.3.1 The Right-Tailed Test 123 4.3.2 The Left-Tailed Test 124 4.3.3 The Two-Tailed Test 125 4.3.4 The Use of JMP 127 4.4 The Probability of a Type II Error and the Power 128 4.4.1 Tests for a Population Mean 130 4.4.2 Tests for a Population Proportion 140 4.4.3 Tests for a Population Variance and Standard Deviation 141 5 Two Hypothesis Tests for the Median of a Population 143 5.1 The Sign Test 144 5.1.1 The Starting Point of the Sign Test 145 5.1.2 Exact p-Values 147 5.1.3 Approximate p-Values Based on the Normal Distribution 148 5.2 The Wilcoxon Signed-Rank Test 151 5.2.1 The Use of Ranks 151 5.2.2 The Starting Point of the Signed-Rank Test 152 5.2.3 Exact p-Values 155 5.2.4 Exact p-Values for Ties 160 5.2.5 Approximate p-Values Based on the Normal Distribution 162 5.2.6 Approximate p-Values Based on the t-Distribution 164 6 Hypothesis Tests for the Distribution of a Population 167 6.1 Testing Probability Distributions 167 6.1.1 Known Parameters 168 6.1.2 Unknown Parameters 170 6.1.3 2-Tests for Qualitative Variables 172 6.2 Testing Probability Densities 181 6.2.1 The Normal Probability Density 182 6.2.2 Other Continuous Densities 195 6.3 Discussion 196 Part Three TWO POPULATIONS 199 7 Independent Versus Paired Samples 201 8 Hypothesis Tests for the Means, Proportions, or Variances of Two Independent Samples 205 8.1 Tests for Two Population Means for Independent Samples 205 8.1.1 The Starting Point 206 2 2 8.1.2 Known Variances 1 and 2 207 2 2 8.1.3 Unknown Variances 1 and 2 212 8.1.4 Confidence Intervals for a Difference in Population Means 222 8.2 A Hypothesis Test for Two Population Proportions 224 x Contents 8.2.1 The Starting Point 225 8.2.2 The Right-Tailed Test 226 8.2.3 The Left-Tailed Test 226 8.2.4 The Two-Tailed Test 226 8.2.5 Generalized Hypothesis Tests 227 8.2.6 The Confidence Interval for a Difference in Population Proportions 228 8.3 A Hypothesis Test for Two Population Variances 229 8.3.1 Fisher’s F-Distribution 230 8.3.2 The F-Test for the Comparison of Two Population Variances 232 8.3.3 The Confidence Interval for a Quotient of Two Population Variances 237 8.4 Hypothesis Tests for Two Independent Samples in JMP 237 8.4.1 Two Population Means 237 8.4.2 Two Population Proportions 244 8.4.3 Two Population Variances 247 9 A Nonparametric Hypothesis Test for the Medians of Two Independent Samples 251 9.1 The Hypotheses Tested 252 9.1.1 The Procedure 252 9.1.2 The Starting Point 253 9.2 Exact p-Values in the Absence of Ties 254 9.2.1 The Right-Tailed Test 256 9.2.2 The Left-Tailed Test 257 9.2.3 The Two-Tailed Test 261 9.3 Exact p-Values in the Presence of Ties 261 9.4 Approximate p-Values 265 9.4.1 The Right-Tailed Test 265 9.4.2 The Left-Tailed Test 266 9.4.3 The Two-Tailed Test 267 10 Hypothesis Tests for the Means of Two Paired Samples 273 10.1 The Hypotheses Tested 273 10.2 The Procedure 274 10.2.1 The Starting Point 274 10.2.2 Known D 277 10.2.3 Unknown D 278 10.3 Examples 280 10.4 The Technical Background 288 10.5 Generalized Hypothesis Tests 289 10.6 A Confidence Interval for a Difference of Two Population Means 290 10.6.1 Known D 290 10.6.2 Unknown D 291 11 Two Nonparametric Hypothesis Tests for Paired Samples 293 11.1 The Sign Test 294 11.1.1 The Hypotheses Tested 294 Contents xi 11.1.2 Practical Implementation 295 11.1.3 JMP 300 11.2