INTRODUCTION TO PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIENTISTS

INTRODUCTION TO PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIENTISTS Third Edition LIMITED WARRANTY AND DISCLAIMER OF LIABILITY Academic Press, (“AP”) and anyone else who has been involved in the creation or production of the accompanying code (“the product”) cannot and not warrant the performance or results that may be obtained by using the product The product is sold “as is” without warranty of merchantability or fitness for any particular purpose AP warrants only that the magnetic diskette(s) on which the code is recorded is free from defects in material and faulty workmanship under the normal use and service for a period of ninety (90) days from the date the product is delivered The purchaser’s sole and exclusive remedy in the event of a defect is expressly limited to either replacement of the diskette(s) or refund of the purchase price, at AP’s sole discretion In no event, whether as a result of breach of contract, warranty, or tort (including negligence), will AP or anyone who 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Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper Copyright © 2004, Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.com.uk You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 0-12-598057-4 (Text) ISBN: 0-12-598059-0 (CD-ROM) For all information on all Academic Press publications visit our Web site at www.academicpress.com Printed in the United States of America 04 05 06 07 08 09 For Elise This Page Intentionally Left Blank CONTENTS Preface xiii Chapter Introduction to Statistics 1.1 1.2 1.3 1.4 1.5 Introduction Data Collection and Descriptive Statistics Inferential Statistics and Probability Models Populations and Samples A Brief History of Statistics Problems Chapter Descriptive Statistics 2.1 Introduction 2.2 Describing Data Sets 2.2.1 Frequency Tables and Graphs 2.2.2 Relative Frequency Tables and Graphs 2.2.3 Grouped Data, Histograms, Ogives, and Stem and Leaf Plots 2.3 Summarizing Data Sets 2.3.1 Sample Mean, Sample Median, and Sample Mode 2.3.2 Sample Variance and Sample Standard Deviation 2.3.3 Sample Percentiles and Box Plots 2.4 Chebyshev’s Inequality 2.5 Normal Data Sets 2.6 Paired Data Sets and the Sample Correlation Coefficient Problems Chapter Elements of Probability 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Introduction Sample Space and Events Venn Diagrams and the Algebra of Events Axioms of Probability Sample Spaces Having Equally Likely Outcomes Conditional Probability Bayes’ Formula vii 1 3 9 10 10 14 17 17 22 24 27 31 33 41 55 55 56 58 59 61 67 70 viii Contents 3.8 Independent Events Problems 76 80 Chapter Random Variables and Expectation 89 89 92 95 101 105 107 111 115 118 121 126 127 130 4.1 Random Variables 4.2 Types of Random Variables 4.3 Jointly Distributed Random Variables 4.3.1 Independent Random Variables *4.3.2 Conditional Distributions 4.4 Expectation 4.5 Properties of the Expected Value 4.5.1 Expected Value of Sums of Random Variables 4.6 Variance 4.7 Covariance and Variance of Sums of Random Variables 4.8 Moment Generating Functions 4.9 Chebyshev’s Inequality and the Weak Law of Large Numbers Problems Chapter Special Random Variables 141 5.1 The Bernoulli and Binomial Random Variables 5.1.1 Computing the Binomial Distribution Function 5.2 The Poisson Random Variable 5.2.1 Computing the Poisson Distribution Function 5.3 The Hypergeometric Random Variable 5.4 The Uniform Random Variable 5.5 Normal Random Variables 5.6 Exponential Random Variables *5.6.1 The Poisson Process *5.7 The Gamma Distribution 5.8 Distributions Arising from the Normal 5.8.1 The Chi-Square Distribution 141 147 148 155 156 160 168 175 179 182 185 185 *5.8.1.1 The Relation Between Chi-Square and Gamma Random Variables 187 5.8.2 The t-Distribution 189 5.8.3 The F-Distribution 191 *5.9 The Logistics Distribution 192 Problems 194 Chapter Distributions of Sampling Statistics 201 6.1 Introduction 201 6.2 The Sample Mean 202 6.3 The Central Limit Theorem 204 Contents ix 6.3.1 Approximate Distribution of the Sample Mean 210 6.3.2 How Large a Sample is Needed? 212 6.4 The Sample Variance 6.5 Sampling Distributions from a Normal Population 6.5.1 Distribution of the Sample Mean 6.5.2 Joint Distribution of X and S 6.6 Sampling from a Finite Population Problems 213 214 215 215 217 221 Chapter Parameter Estimation 229 7.1 Introduction 7.2 Maximum Likelihood Estimators *7.2.1 Estimating Life Distributions 7.3 Interval Estimates 229 230 238 240 7.3.1 Confidence Interval for a Normal Mean When the Variance is Unknown 246 7.3.2 Confidence Intervals for the Variances of a Normal Distribution 251 7.4 Estimating the Difference in Means of Two Normal Populations 7.5 Approximate Confidence Interval for the Mean of a Bernoulli Random Variable *7.6 Confidence Interval of the Mean of the Exponential Distribution *7.7 Evaluating a Point Estimator *7.8 The Bayes Estimator Problems 253 260 265 266 272 277 Chapter Hypothesis Testing 291 8.1 Introduction 8.2 Significance Levels 8.3 Tests Concerning the Mean of a Normal Population 8.3.1 Case of Known Variance 8.3.2 Case of Unknown Variance: The t-Test 8.4 Testing the Equality of Means of Two Normal Populations 8.4.1 Case of Known Variances 8.4.2 Case of Unknown Variances 8.4.3 Case of Unknown and Unequal Variances 8.4.4 The Paired t-Test 8.5 Hypothesis Tests Concerning the Variance of a Normal Population 291 292 293 293 305 312 312 314 318 319 321 8.5.1 Testing for the Equality of Variances of Two Normal Populations 322 8.6 Hypothesis Tests in Bernoulli Populations 323 8.6.1 Testing the Equality of Parameters in Two Bernoulli Populations 327 610 Chapter 14*: Life Testing (b) Use part (a) to show that E [U (i)] = i/(n + 1) [Hint: To evaluate the resulting integral, use the fact that the density in part (a) must integrate to 1.] (c) Use part (b) and Problem 28a to conclude that E [F (X(i) )] = i/(n + 1) 30 If U is uniformly distributed on (0, 1), show that − log U has an exponential distribution with mean Now use Equation 14.3.7 and the results of the previous problems to establish Equation 14.5.7 APPENDIX OF TABLES A1: A2: A3: A4: A5: Standard Normal Distribution Function Probabilities for Chi-Square Random Variables Probabilities for t-Random Variables Probabilities for F-Random Variables ANOVA Multiple Comparison Constants 611 612 TABLE A1 Appendix of Tables Standard Normal Distribution Function: x (x) = √ e −y /2 dy 2π −∞ x 00 01 02 03 04 05 06 07 08 09 5000 5398 5793 6179 6554 6915 7257 7580 7881 8159 5040 5438 5832 6217 6591 6950 7291 7611 7910 8186 5080 5478 5871 6255 6628 6985 7324 7642 7939 8212 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 5160 5557 5948 6331 6700 7054 7389 7704 7995 8264 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 5239 5636 6026 6406 6772 7123 7454 7764 8051 8315 5279 5675 6064 6443 6808 7157 7486 7794 8078 8340 5319 5714 6103 6480 6844 7190 7517 7823 8106 8365 5359 5753 6141 6517 6879 7224 7549 7852 8133 8389 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 8438 8665 8869 9049 9207 9345 9463 9564 9649 9719 8461 8686 8888 9066 9222 9357 9474 9573 9656 9726 8485 8708 8907 9082 9236 9370 9484 9582 9664 9732 8508 8729 8925 9099 9251 9382 9495 9591 9671 9738 8531 8749 8944 9115 9265 9394 9505 9599 9678 9744 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 8577 8790 8980 9147 9292 9418 9525 9616 9693 9756 8599 8810 8997 9162 9306 9429 9535 9625 9699 9761 8621 8830 9015 9177 9319 9441 9545 9633 9706 9767 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 9772 9821 9861 9893 9918 9938 9953 9965 9974 9981 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 9783 9830 9868 9898 9922 9941 9956 9967 9976 9982 9788 9834 9871 9901 9925 9943 9957 9968 9977 9983 9793 9838 9875 9904 9927 9945 9959 9969 9977 9984 9798 9842 9878 9906 9929 9946 9960 9970 9978 9984 9803 9846 9881 9909 9931 9948 9961 9971 9979 9985 9808 9850 9884 9911 9932 9949 9962 9972 9979 9985 9812 9854 9887 9913 9934 9951 9963 9973 9980 9986 9817 9857 9890 9916 9936 9952 9964 9974 9981 9986 3.0 3.1 3.2 3.3 3.4 9987 9990 9993 9995 9997 9987 9991 9993 9995 9997 9987 9991 9994 9995 9997 9988 9991 9994 9996 9997 9988 9992 9994 9996 9997 9989 9992 9994 9996 9997 9989 9992 9994 9996 9997 9989 9992 9995 9996 9997 9990 9993 9995 9996 9997 9990 9993 9995 9997 9998 Appendix of Tables TABLE A2 613 Values of xα,n n α = 995 0000393 0100 0717 207 412 α = 99 α = 975 α = 95 α = 05 α = 025 α = 01 α = 005 000157 0201 115 297 554 000982 0506 216 484 831 00393 103 352 711 1.145 3.841 5.991 7.815 9.488 11.070 5.024 7.378 9.348 11.143 12.832 6.635 9.210 11.345 13.277 13.086 7.879 10.597 12.838 14.860 16.750 10 676 989 1.344 1.735 2.156 872 1.239 1.646 2.088 2.558 1.237 1.690 2.180 2.700 3.247 1.635 2.167 2.733 3.325 3.940 12.592 14.067 15.507 16.919 18.307 14.449 16.013 17.535 19.023 20.483 16.812 18.475 20.090 21.666 23.209 18.548 20.278 21.955 23.589 25.188 11 12 13 14 15 2.603 3.074 3.565 4.075 4.601 3.053 3.571 4.107 4.660 5.229 3.816 4.404 5.009 5.629 6.262 4.575 5.226 5.892 6.571 7.261 19.675 21.026 22.362 23.685 24.996 21.920 23.337 24.736 26.119 27.488 24.725 26.217 27.688 29.141 30.578 26.757 28.300 29.819 31.319 32.801 16 17 18 19 20 5.142 5.697 6.265 6.844 7.434 5.812 6.408 7.015 7.633 8.260 6.908 7.564 8.231 8.907 9.591 7.962 8.672 9.390 10.117 10.851 26.296 27.587 28.869 30.144 31.410 28.845 30.191 31.526 32.852 34.170 32.000 33.409 34.805 36.191 37.566 34.267 35.718 37.156 38.582 39.997 21 8.034 22 8.643 23 9.260 24 9.886 25 10.520 8.897 9.542 10.196 10.856 11.524 10.283 10.982 11.689 12.401 13.120 11.591 12.338 13.091 13.484 14.611 32.671 33.924 35.172 36.415 37.652 35.479 36.781 38.076 39.364 40.646 38.932 40.289 41.638 42.980 44.314 41.401 42.796 44.181 45.558 46.928 26 27 28 29 30 12.198 12.879 13.565 14.256 14.953 13.844 14.573 15.308 16.047 16.791 15.379 16.151 16.928 17.708 18.493 38.885 40.113 41.337 42.557 43.773 41.923 43.194 44.461 45.772 46.979 45.642 46.963 48.278 49.588 50.892 48.290 49.645 50.993 52.336 53.672 11.160 11.808 12.461 13.121 13.787 Other Chi-Square Probabilities: = 4.2 P{x < 14.3} = 425 x.9,9 16 < 17.1875} = 8976 P{x11 614 Appendix of Tables TABLE A3 Values of tα,n n α = 10 α = 05 α = 025 α = 01 α = 005 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.706 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.474 3.365 63.657 9.925 5.841 4.604 4.032 10 1.440 1.415 1.397 1.383 1.372 1.943 1.895 1.860 1.833 1.812 2.447 2.365 2.306 2.262 2.228 3.143 2.998 2.896 2.821 2.764 3.707 3.499 3.355 3.250 3.169 11 12 13 14 15 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 16 17 18 19 20 1.337 1.333 1.330 1.328 1.325 1.746 1.740 1.734 1.729 1.725 2.120 2.110 2.101 2.093 2.086 2.583 2.567 2.552 2.539 2.528 2.921 2.898 2.878 2.861 2.845 21 22 23 24 25 1.323 1.321 1.319 1.318 1.316 1.721 1.717 1.714 1.711 1.708 2.080 2.074 2.069 2.064 2.060 2.518 2.508 2.500 2.492 2.485 2.831 2.819 2.807 2.797 2.787 26 27 28 29 1.315 1.314 1.313 1.311 1.706 1.703 1.701 1.699 2.056 2.052 2.048 2.045 2.479 2.473 2.467 2.462 2.779 2.771 2.763 2.756 ∞ 1.282 1.645 1.960 2.326 2.576 Other t Probabilities: P{T8 < 2.541} = 9825 P{T8 < 2.7} = 9864 1.66} = 94 P{T12 < 2.8} = 984 P{T11 < 7635} = 77 P{T11 < 934} = 81 P{T11 < Appendix of Tables 615 Values of F.05,n,m TABLE A4 m = Degrees of Freedom for Denominator n = Degrees of Freedom for Numerator 5 161 18.50 10.10 7.71 6.61 200 19.00 9.55 6.94 5.79 216 19.20 9.28 6.59 5.41 225 19.20 9.12 6.39 5.19 230 19.30 9.01 6.26 5.05 10 5.99 5.59 5.32 5.12 4.96 5.14 4.74 4.46 4.26 4.10 4.76 4.35 4.07 3.86 3.71 4.53 4.12 3.84 3.63 3.48 4.39 3.97 3.69 3.48 3.33 11 12 13 14 15 4.84 4.75 4.67 4.60 4.54 3.98 3.89 3.81 3.74 3.68 3.59 3.49 3.41 3.34 3.29 3.36 3.26 3.18 3.11 3.06 3.20 3.11 3.03 2.96 2.90 16 17 18 19 20 4.49 3.45 4.41 4.38 4.35 3.63 3.59 3.55 3.52 3.49 3.24 3.20 3.16 3.13 3.10 3.01 2.96 2.93 2.90 2.87 2.85 2.81 2.77 2.74 2.71 21 22 23 24 25 4.32 4.30 4.28 4.26 4.24 3.47 3.44 3.42 3.40 3.39 3.07 3.05 3.03 3.01 2.99 2.84 2.82 2.80 2.78 2.76 2.68 2.66 2.64 2.62 2.60 30 40 60 120 4.17 4.08 4.00 3.92 3.32 3.23 3.15 3.07 2.92 2.84 2.76 2.68 2.69 2.61 2.53 2.45 2.53 2.45 2.37 2.29 ∞ 3.84 3.00 2.60 2.37 2.21 Other F Probabilities: F.1,7,5 = 337 P{F7.7 < 1.376} = 316 P{F20,14 < 2.461} = 911 P{F9,4 < 5} = 1782 616 TABLE A5 Appendix of Tables Values of C(m, d, α) m d α 10 11 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 05 01 3.64 5.70 3.46 5.24 3.34 4.95 3.26 4.75 3.20 4.60 3.15 4.48 3.11 4.39 3.08 4.32 3.06 4.26 3.03 4.21 3.01 4.17 3.00 4.13 2.98 4.10 2.97 4.07 2.96 4.05 2.95 4.02 2.92 3.96 2.89 3.89 2.86 3.82 2.83 3.76 2.80 3.70 2.77 3.64 4.60 6.98 4.34 6.33 4.16 5.92 4.04 5.64 3.95 5.43 3.88 5.27 3.82 5.15 3.77 5.05 3.73 4.96 3.70 4.89 3.67 4.84 3.65 4.79 3.63 4.74 3.61 4.70 3.59 4.67 3.58 4.64 3.53 4.55 3.49 4.45 3.44 4.37 3.40 4.28 3.36 4.20 3.31 4.12 5.22 7.80 4.90 7.03 4.68 6.54 4.53 6.20 4.41 5.96 4.33 5.77 4.26 5.62 4.20 5.50 4.15 5.40 4.11 5.32 4.08 5.25 4.05 5.19 4.02 5.14 4.00 5.09 3.98 5.05 3.96 5.02 3.90 4.91 3.85 4.80 3.79 4.70 3.74 4.59 3.68 4.50 3.63 4.40 5.67 8.42 5.30 7.56 5.06 7.01 4.89 6.62 4.76 6.35 4.65 6.14 4.57 5.97 4.51 5.84 4.45 5.73 4.41 5.63 4.37 5.56 4.33 5.49 4.30 5.43 4.28 5.38 4.25 5.33 4.23 5.29 4.17 5.17 4.10 5.05 4.04 4.93 3.98 4.82 3.92 4.71 3.86 4.60 6.03 8.91 5.63 7.97 5.36 7.37 5.17 6.96 5.02 6.66 4.91 6.43 4.82 6.25 4.75 6.10 4.69 5.98 4.64 5.88 4.59 5.80 4.56 5.72 4.52 5.66 4.49 5.60 4.47 5.55 4.45 5.51 4.37 5.37 4.30 5.24 4.23 5.11 4.16 4.99 4.10 4.87 4.03 4.76 6.33 9.32 5.90 8.32 5.61 7.68 5.40 7.24 5.24 6.91 5.12 6.67 5.03 6.48 4.95 6.32 4.88 6.19 4.83 6.08 4.78 5.99 4.74 5.92 4.70 5.85 4.67 5.79 4.65 5.73 4.62 5.69 4.54 5.54 4.46 5.40 4.39 5.26 4.31 5.13 4.24 5.01 4.17 4.88 6.58 9.67 6.12 8.61 5.82 7.94 5.60 7.47 5.43 7.13 5.30 6.87 5.20 6.67 5.12 6.51 5.05 6.37 4.99 6.26 4.94 6.16 4.90 6.08 4.86 6.01 4.82 5.94 4.79 5.89 4.77 5.84 4.68 5.69 4.60 5.54 4.52 5.39 4.44 5.25 4.36 5.12 4.29 4.99 6.80 9.97 6.32 8.87 6.00 8.17 5.77 7.68 5.59 7.33 5.46 7.05 5.35 6.84 5.27 6.67 5.19 6.53 5.13 6.41 5.08 6.31 5.03 6.22 4.99 6.15 4.96 6.08 4.92 6.02 4.90 5.97 4.81 5.81 4.72 5.65 4.63 5.50 4.55 5.36 4.47 5.21 4.39 5.08 6.99 10.24 6.49 9.10 6.16 8.37 5.92 7.86 5.74 7.49 5.60 7.21 5.49 6.99 5.39 6.81 5.32 6.67 5.25 6.54 5.20 6.44 5.15 6.35 5.11 6.27 5.07 6.20 5.04 6.14 5.01 6.09 4.92 5.92 4.82 5.76 4.73 5.60 4.65 5.45 4.56 5.30 4.47 5.16 7.17 10.48 6.65 9.30 6.30 8.55 6.05 8.03 5.87 7.65 5.72 7.36 5.61 7.13 5.51 6.94 5.43 6.79 5.36 6.66 5.31 6.55 5.26 6.46 5.21 6.38 5.17 6.31 5.14 6.25 5.11 6.19 5.01 6.02 4.92 5.85 4.82 5.69 4.73 5.53 4.64 5.37 4.55 5.23 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 ∞ Index A B Analysis of variance (ANOVA) applications, 439–440 multiple comparison constants, 616 one-way, 440 between samples sum of squares, 445–446, 453 multiple comparisons of sample means, 450–452 null hypothesis, 442, 445–446 sum of squares identity, 447–450 unequal sample sizes, 452–453 within samples sum of squares, 443–445, 452–453 overview, 440–442 two-way, 440, 454–457 column sum of squares, 461 error sum of squares, 459, 465 grand mean, 456, 463 hypothesis testing, 458–462 null hypothesis, 464, 467–468 parameter estimation, 458–459 row and column interaction, 463–468 row sum of squares, 460–461 Approximately normal data set, 31 Assignable cause, 545 Attribute, quality control, 545 Axioms of probability, 59–61 Bar graph, 10 Basic principle of counting generalized, 63 proof, 62–63 Bayes, T., 75 Bayes estimator, 272–274 Bernoulli random variables, 273–274 normal mean, 274–275 life testing, 596–598 Bayes formula, 70–76 Behrens-Fisher problem, 319 Bernoulli, J., 5, 141 Bernoulli random variable, 141, 157–158 approximate confidence interval for mean of distribution, 260–264 Bayes estimator, 273–274 testing equality of parameters in two Bernoulli populations, 327–329 Beta distribution, 274 Between samples sum of squares, 445–446, 453 Bias of an estimator, 267, 272 Bimodal data set, 33–34 Binomial distribution function, 147–148 Binomial random variable definition, 141 hypergeometric random variable relationship, 159–160, 219–220 probability calculations, 142–146 probability mass function, 142 617 618 Binomial random variable (continued ) testing equality of parameters in two Bernoulli populations, 327–329 Binomial theorem, 142 Box plot, 27 C Central limit theorem, 204–206, 209–210 approximate distribution of sample mean, 210–212 sample size requirements, 212–213 Chance variation, 545 Chebyshev’s inequality for data sets definition, 9, 27–28 one-sided Chebyshev inequality, 29–30 proof, 28 probabilities, 127–129 Chi-square distribution definition, 185–186 probabilities for random variables, 613 relation between chi-square and gamma random variables, 187–189 sum of squares of residuals in linear regression, 358–359 summation of random variables, 216 Class interval, 14–15 Coefficient of determination, 376–378 Coefficient of multiple determination, 405 Column sum of squares, 461 Combinations of objects, 65 Composite hypothesis, 292 Conditional distribution, 105–107 Conditional probability, 67–70 Confidence interval estimators, 241–242 Bernoulli mean, 260–264 difference of two normal means, 253–260 exponential mean, 265–266 interpretation, 245–246 normal mean with unknown variance, 246–249, 251 one-sided lower, 242–243, 249 one-sided upper, 242–243, 248 Index regression parameters a, 370 b, 365–366 mean response, 372–373 two-sided confidence interval, 242, 244–245 variances of a normal distribution, 251–253 Contingency tables with fixed marginal totals, 499–504 tests of independence, 495–499 Continuous random variable, 91, 93 Control charts cumulative sum charts, 571–573 estimation of mean and variance, 549–551 exponentially weighted moving-average charts, 565–570 for fraction defective, 557–559 lower control limit, 547–548, 552–553, 555–559, 562 moving-average charts, 563–565 for number of defects, 559–562 S-charts, 554 upper control limit, 547–548, 552–553, 555–559, 562 X -charts, 546–554 Correlation coefficient, see Sample correlation coefficient Covariance definition, 121–122 multiple linear regression, 399–401 properties, 122–123 sums of random variables, 125–126 Cumulative sum control charts, 571–573 D de Moivre, A., 168 DeMorgan’s laws, 59 Dependent variable, 351 Descriptive statistics, 1–2, Discrete random variable, 91–92 Distribution function, 91–93 Doll, R., 17 Double-blind test, 164 Index E Effect of column, 464 Effect of row, 464 Empirical rule, 32–33 Entropy, 109–111 Error sum of squares, 459, 465 Estimate, 230 Estimated regression line, 354 Estimator, 230 Event algebraic operations, 58–59 axioms of probability, 59–61 complement, 57 definition, 56 independent events, 76–80 intersection of events, 57 mutually exclusive events, 57 union of events, 57 Expectation, see Expected value Expected value calculation, 107–109 definition, 107 expectation of a function of a random variable, 113–115 nomenclature, 115 properties, 111–113 sums of random variables, 115–118 Exponentially weighted moving-average control charts, 565–570 Exponential random variable confidence interval for mean of distribution, 260–264 definition, 175–176 memoryless property, 176–178 moment generating function, 176 Poisson process, 179–181 properties, 176–179 F Failure rate, 239, 581 functions, 581–584 Finite population sampling, 217–221 First moment, 115 Fisher, R A., Fisher-Irwin test, 328–329 619 F-random variable distribution, 191–192 probabilities for, 615 Frequency interpretation of probability, 55 Frequency polygon, 10 Frequency table, 10 G Galton, F., 6, 366 Gamma distribution definition, 182 moment generating function of gamma random variable, 183 properties of gamma random variables, 183–185 relation between chi-square and gamma random variables, 187–189 Gamma function, 183 Gauss, K F., Generalized basic principle of counting, 63 Goodness of fit tests, 483 critical region determination by simulation, 490–493 Kolmogorov–Smirnov test, 504–508 specified parameters, 484–489 tests of independence in contingency tables, 495–499 with fixed marginal totals, 499–504 unspecified parameters, 493–495 Gosset, W S., Grand mean, 456, 463 Graphs bar graph, 10, 16 frequency polygon, 10 line graph, 10 relative frequency graph, 10, 12 Graunt, J., 4–5 H Halley, E., Hazard rate, see Failure rate Herbst, A., 329 Hill, A B., 17 620 Histogram bimodal data set, 33–34 definition, 16 normal data set, 31 Hotel, D G., 21 Hypergeometric random variable, 156–157 binomial random variable relationship, 159–160, 219–220 mean, 157 variance, 157–158 Hypothesis test, see Statistical hypothesis test I Independent events, 76–80 Independent random variables, 101–105 Independent variable, 351 Indicator random variable, 90–91 expectation, 109 variance, 120 Inferential statistics, 2–3 Information theory, entropy, 108 Interaction of row and column in analysis of variance, 464 Interval estimates, 240 J Joint distribution, sample mean and sample variance in normal population, 215–217 Jointly continous random variables, 99 Jointly distributed random variables, 95–101 Joint probability density function, 99–100 Joint probability mass function, 96 K Kolmogorov’s law of fragmentation, 237–238 Kolmogorov–Smirnov goodness of fit test, 504–508 Kolmogorov–Smirnov test statistic, 504–507 L Laplace, P., Least squares estimators in linear regression distribution of estimators, 355–362 Index estimated regression line, 354 mean and variance computation, 356–357 multiple linear regression, 394–405 normal equations, 353–354 notation, 360 sum of squared differences, 353 weighted least squares, 384–390 Left-end inclusion convention, 15 Life testing exponential distribution Bayesian appproach, 596–598 sequential testing, 590–594 simultaneus testing, 584–590, 594 hazard rate functions, 581–584 maximum likelihood estimator of life distributions, 238–240 Likelihood function, 230 parameter estimation by least squares, 602–604 two-sample problem, 598–600 Weibull distribution, 600–602 Linear regression equation, 351–352 Linear transformation, 381–384 Line graph, 10 Logistic regression model, 410–413 Logistics regression function, 410 Logistics distribution, 192–193 Logit, 411 Lower control limit, 547–548, 552–553, 555–559, 562 M Mann-Whitney test, 525 Marginal probability mass function, 98 Markov’s inequality, 127–129 Maximum likelihood estimator of Bernoulli parameter, 231–233 definition, 230–231 Kolmogorov’s law of fragmentation, 237–238 of life distributions, 238–240 of normal population, 236–238 of Poisson parameter, 234–235 of uniform distribution, 238 Mean, see Sample mean Index Mean response confidence interval estimator, 372–373, 405–408 prediction interval of future response, 373–375, 410 statistical inferences, 371–372 Mean square error, 266–271 Median, see Sample median Memoryless property, 176–178 Modal value, 22 Mode, see Sample mode Mode of a density, 276–277 Moment generating function, 126–127 exponential random variable, 176 gamma random variable, 183 normal random variable, 169–170, 173 Poisson random variable, 149–150, 154 Monte Carlo simulation, 251 Moving-average control charts, 563–565 Multiple linear regression, 394–405 Multiple regression equation, 352 Multivariate normal disttribution, 398 N Negatively correlated, 36 Newton, I., Neyman, J., Nonparametric hypothesis tests definition, 515 rank sum test classical approximation, 529–531 null hypothesis, 526 simulation, 531–533 T statistic, 525–529 runs test for randomness, 533–536 signed rank test, 519–525 sign test, 515–519 Nonparametric interference problem, 202 Normal data set approximately normal data set, 31 definition, 31 empirical rule, 32–33 histogram, 31 Normal density, 275 Normal distribution, 168–170 621 Normal equations, 395 Normal prior distribution, 275–277 Normal random variable definition, 168 moment generating function, 169–170, 173 probability calculations, 174–175 summation, 173 Null hypothesis, 292 analysis of variance one-way, 442, 445–446 two-way, 464, 467–468 Bernoulli populations, 323–330 equality of normal variances, 321–323 equality of two normal means known variances, 312–314 paired t-test, 319–320 unknown and unequal variances, 318 unknown variances, 314–318 goodness of fit tests, 484–485, 494, 496, 502, 504 normal population mean with known variance, 293–305 one-sided tests, 300–305 Poisson distribution mean, 330–333 rank sum test, 526 regression parameter b, 363–365 signed rank test, 521 sign test, 515, 517, 519 O Observational study, 329 Ogive, 16 One-way analysis of variance, see Analysis of variance Operating characteristic curve, 297 Order statistics, 586 Out of control process, 545 Overlook probabilities, 76 P Paired data sets, 33–36 Paired t-test, 319–320 Parametric interference problem, 201–202 Pearson, E., Pearson, K., 6, 367, 490 622 Permutation, 63 Pie chart, 12 Point estimator, 240–241 evaluation, 266–272 mean square error, 266–271 unbiased estimator, 267 Poisson, S D., 148 Poisson distribution function, 155–156 Poisson process, 179–181 Poisson random variable applications, 150–153 definition, 148 moment generating function, 149–150, 154 square root, 389–390 tests concerning Poisson distribution mean, 330–333 Polynomial regression, 391–394 Pooled estimator, 259, 315 Population, 3, 201 Population mean, 202 Population variance, 202 Positively correlated, 36 Poskanzer, D., 329 Posterior density function, 273, 276 Power function, 298 Prediction interval, future response in regression, 373–375, 410 Prior distribution, 272, 275–277 Probability axioms, 59–61 conditional, 67–70 frequency interpretation, 55 subjective interpretation, 55 Probability density function joint probability density function, 99–100 random variable, 93–95 sample means, 203 Probability mass function binomial random variable, 142 joint probability mass function, 96 marginal probability mass function, 98 random variable, 92 Probit model, 412 Pseudo random number, 251 p-value, 296, 303–304, 309, 311 Index Q Quadratic regression equation, 393 Quality control, see Control charts Quartiles, 25–27 R Randomness, runs test, 533–536 Random number, 163 Random sample, 217 Random variable Bernoulli random variable, 141 binomial random variable, 141–148 chi-square distribution, 185–187 conditional distributions, 105–107 continuous random variable, 91, 93 covariance definition, 121–122 properties, 122–123 sums of random variables, 125–126 definition, 89–90 discrete random variable, 91–92 distribution function, 91–93 entropy, 109–111 expectation, 107–118 exponential random variables, 175–181 F-distribution, 191–192 gamma distribution, 182–185 hypergeometric random variable, 156–160 independent random variables, 101–105 indicator random variable, 90–91 jointly distributed random variables, 95–101 logistics distribution, 192–193 moment generating functions, 126–127 normal random variables, 168–175 Poisson random variable, 148–156 probability density function, 93–95 probability mass function, 92 sums of random variables, expected value, 115–118 t-distribution, 189–191 uniform random variable, 160–168 variance definition, 118–120 standard deviation, 121, 126 sums of random variables, 123–125 Index Range of data, 27 Rank sum test classical approximation, 529–531 null hypothesis, 526 simulation, 531–533 T statistic, 525–529 Rayleigh density function, 583 Regression coefficients definition, 352 statistical inferences concerning regression parameters a, 370 b, 362–370 mean response, 371–373 prediction intervals of future response, 373–375 Regression fallacy, 370 Regression to the mean, 366–370 Relative frequency, 10, 12, 15 Residuals in regression, 358 model assessment, 378–380 standardized residuals, 379 sum of squares chi-square distribution, 358–359 computational identity, 360–362 multiple linear regression, 397, 402–403 Robust test, 305 Row sum of squares, 460–461 Runs test, 533–536 S Sample, 3, 201–202 Sample correlation coefficient coefficient of determination relationship, 378 definition, 36 positive versus negative correlations, 36 properties, 37–40 Sample mean analysis of variance for multiple comparisons of sample means, 450–452 approximate distribution, 210–212 definition, 17, 19, 202–203 distribution from a normal population, 215 623 joint distribution with sample variance, 215–217 probability density function, 203 Sample median, 20–21 Sample mode, 21 Sample percentile definition, 25 quartiles, 25–27 Sample space definition, 56 spaces having equally likely outcomes, 61–67 Sample standard deviation, 24, 213 Sample variance algebraic identity for computation, 23–24 definition, 22–23, 213–214 joint distribution with sample mean, 215–217 Scatter diagram, 34, 352 S-control charts, 554 Sequence of interarrival times, 181 Shockley equation, 162 Signed rank test, 519–525 Significance level, 293, 306, 309 Sign test, 515–519 Simple hypothesis, 292 Simple regression equation, 352 Skewed data set, 31 Standard deviation, see Sample standard deviation; Variance Standard normal distribution function, 170–171, 175, 612 Standardized residuals, 379 Statistic, 202 Statistical hypothesis test Bernoulli populations, 323–330 composite hypothesis, 292 definition, 291 equality of normal variances, 321–323 equality of two normal means known variances, 312–314 paired t-test, 319–320 unknown and unequal variances, 318 unknown variances, 314–318 level of significance, 293 normal population mean with known variance, 293–305 624 Statistical hypothesis test (continued ) null hypothesis, 292 one-sided tests, 300–305 Poisson distribution mean, 330–333 power function, 298 p-value, 296, 303–304 regression parameter b, 363–365 robustness, 305 simple hypothesis, 292 t-test, 305–311 Statistics definition, 1, descriptive, 1–2, historical perspective, 3–7 inferential, 2–3 summarizing, 17 Stem and leaf plot, 16–17 Subjective interpretation of probability, 55 Sum of squares identity, 447–450 Survival rate, 239–240 T Total time-on test statistic, 586 t-random variable distribution, 189–191 probabilities for, 614 Tree diagram, 166 T statistic, 306–307, 310, 368, 445–446, 484–485, 489, 525 t-test, 305–306 level of significance, 306, 309 Index p-value, 307–310 two-sided tests, 307–311 Two-factor analysis of variance, see Analysis of variance Type I error, 292 Type II error, 292 U Ulfelder, H., 329 Unbiased estimator, 267, 271, 357–358, 398 Uniform distribution, 166–168 Uniform random variable, 160–168 Unit normal distribution, 170 Upper control limit, 547–548, 552–553, 555–559, 562 V Variance, see also Sample variance definition, 118–120 standard deviation, 121, 126 sums of random variables, 123–125 Venn diagram, 58 W Weak law of large numbers, 129–130 Weibull distribution, 600–602 Weighted average, 19 Weighted least squares, 384–390 Wilcoxon test, 525 Within samples sum of squares, 443–445, 452–453 [...]... ideas of statistics are everywhere Descriptive statistics are featured in every newspaper and magazine Statistical inference has become indispensable to public health and medical research, to engineering and scientific studies, to marketing and quality control, to education, to accounting, to economics, to meteorological forecasting, to polling and surveys, to sports, to insurance, to gambling, and to all... most often used by practicing engineers and scientists This book has been written for an introductory course in statistics, or in probability and statistics, for students in engineering, computer science, mathematics, statistics, and the natural sciences As such it assumes knowledge of elementary calculus ORGANIZATION AND COVERAGE Chapter 1 presents a brief introduction to statistics, presenting its two... description and summarization of data, is called descriptive statistics 1.3 INFERENTIAL STATISTICS AND PROBABILITY MODELS After the preceding experiment is completed and the data are described and summarized, we hope to be able to draw a conclusion about which teaching method is superior This part of statistics, concerned with the drawing of conclusions, is called inferential statistics To be able to draw... data, it is necessary to have an understanding of the data’s origination For instance, it is often assumed that the data constitute a “random sample” from some population To understand exactly what this means and what its consequences are for relating properties of the sample data to properties of the entire population, it is necessary to have some understanding of probability, and that is the subject... descriptive and inferential statistics, and a short history of the subject and some of the people whose early work provided a foundation for work done today The subject matter of descriptive statistics is then considered in Chapter 2 Graphs and tables that describe a data set are presented in this chapter, as are quantities that are used to summarize certain of the key properties of the data set To be able to. .. Digest would work better today? 3 A researcher is trying to discover the average age at death for people in the United States today To obtain data, the obituary columns of the New York Times are read for 30 days, and the ages at death of people in the United States are noted Do you think this approach will lead to a representative sample? 8 Chapter 1: Introduction to Statistics 4 To determine the proportion... throughout the whole of Science and Technology (E Pearson, 1936) Statistics is the name for that science and art which deals with uncertain inferences — which uses numbers to find out something about nature and experience (Weaver, 1952) Statistics has become known in the 20th century as the mathematical tool for analyzing experimental and observational data (Porter, 1986) Statistics is the art of learning... that one wants to make of the data For instance, suppose that an instructor is interested in determining which of two different methods for teaching computer programming to beginners is most effective To study this question, the instructor might divide the students into two groups, and use a different teaching method for each group At the end of the class the students can be tested and the scores of... statistics, derived from the word state, was used to refer to a collection of facts of interest to the state The idea of 4 Chapter 1: Introduction to Statistics collecting data spread from Italy to the other countries of Western Europe Indeed, by the first half of the 16th century it was common for European governments to require parishes to register births, marriages, and deaths Because of poor public health... introduces the idea of a probability experiment, explains the concept of the probability of an event, and presents the axioms of probability Our study of probability is continued in Chapter 4, which deals with the important concepts of random variables and expectation, and in Chapter 5, which considers some special types of random variables that often occur in applications Such random variables as the