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What’s on the CD-ROM?Included on the accompanying CD-ROM: a fully searchable eBook version of the text in Adobe pdf form data sets to accompany the examples in the text in the “Extras” folder, useful statistical software tools developed by theStatistical Engineering Division, National Institute of Science andTechnology (NIST). Once again, you are cautioned not to apply any tech nique blindly without first understanding its assumptions, limitations, andarea of application.Refer to the Read-Me file on the CD-ROM for more detailed information onthese files and applications.xiii

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List of SymbolsA or A′A BA BB AE(X)f(x)fiinCn rPn rpp̂p(xi)Pr [.]qQ(f )ss2sc2sy2 xtcomplement of Aintersection of A and Bunion of A and Bconditional probabilityexpectation of random variable Xprobability density functionfrequency of result xiorder numbernumber of trialsnumber of combinations of n items taken r at a timenumber of permutations of n items taken r at a timeprobability of “success” in a single trialestimated proportionprobability of result xiprobability of stated outcome or eventprobability of “no success” in a single trialquantile larger than a fraction f of a distributionestimate of standard deviation from a sampleestimate of variance from a samplecombined or pooled estimate of varianceestimated variance around a regression lineinterval of time or space. Also the independent variable of thet-distribution.X (capital letter) a random variablex (lower case)a particular value of a random variablearithmetic mean or mean of a samplexzratio between (x – µ) and σ for the normal distributionαregression coefficientβregression coefficientλmean rate of occurrence per unit time or spaceµmean of a populationσstandard deviation of populationstandard error of the meanσxvariance of populationσ2xv

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CHAPTER1Introduction:Probability and StatisticsProbability and statistics are concerned with events which occur by chance. Examplesinclude occurrence of accidents, errors of measurements, production of defective andnondefective items from a production line, and various games of chance, such asdrawing a card from a well-mixed deck, flipping a coin, or throwing a symmetricalsix-sided die. In each case we may have some knowledge of the likelihood of variouspossible results, but we cannot predict with any certainty the outcome of any particu lar trial. Probability and statistics are used throughout engineering. In electricalengineering, signals and noise are analyzed by means of probability theory. Civil,mechanical, and industrial engineers use statistics and probability to test and accountfor variations in materials and goods. Chemical engineers use probability and statis tics to assess experimental data and control and improve chemical processes. It isessential for today’s engineer to master these tools.1.1 Some Important Terms(a) Probability is an area of study which involves predicting the relative likeli hood of various outcomes. It is a mathematical area which has developedover the past three or four centuries. One of the early uses was to calculatethe odds of various gambling games. Its usefulness for describing errors ofscientific and engineering measurements was soon realized. Engineers studyprobability for its many practical uses, ranging from quality control andquality assurance to communication theory in electrical engineering. Engi neering measurements are often analyzed using statistics, as we shall seelater in this book, and a good knowledge of probability is needed in order tounderstand statistics.(b) Statistics is a word with a variety of meanings. To the man in the street it mostoften means simply a collection of numbers, such as the number of peopleliving in a country or city, a stock exchange index, or the rate of inflation.These all come under the heading of descriptive statistics, in which items arecounted or measured and the results are combined in various ways to giveuseful results. That type of statistics certainly has its uses in engineering, and1

Chapter 1we will deal with it later, but another type of statistics will engage ourattention in this book to a much greater extent. That is inferential statistics orstatistical inference. For example, it is often not practical to measure all theitems produced by a process. Instead, we very frequently take a sample andmeasure the relevant quantity on each member of the sample. We infersomething about all the items of interest from our knowledge of the sample.A particular characteristic of all the items we are interested in constitutes apopulation. Measurements of the diameter of all possible bolts as they comeoff a production process would make up a particular population. A sample isa chosen part of the population in question, say the measured diameters oftwelve bolts chosen to be representative of all the bolts made under certainconditions. We need to know how reliable is the information inferred aboutthe population on the basis of our measurements of the sample. Perhaps wecan say that “nineteen times out of twenty” the error will be less than acertain stated limit.(c) Chance is a necessary part of any process to be described by probabilityor statistics. Sometimes that element of chance is due partly or even perhapsentirely to our lack of knowledge of the details of the process. For example,if we had complete knowledge of the composition of every part of the rawmaterials used to make bolts, and of the physical processes and conditions intheir manufacture, in principle we could predict the diameter of each bolt.But in practice we generally lack that complete knowledge, so the diameterof the next bolt to be produced is an unknown quantity described by arandom variation. Under these conditions the distribution of diameters can bedescribed by probability and statistics. If we want to improve the quality ofthose bolts and to make them more uniform, we will have to look into thecauses of the variation and make changes in the raw materials or the produc tion process. But even after that, there will very likely be a random variationin diameter that can be described statistically.Relations which involve chance are called probabilistic or stochastic rela tions. These are contrasted with deterministic relations, in which there is noelement of chance. For example, Ohm’s Law and Newton’s Second Lawinvolve no element of chance, so they are deterministic. However, measure ments based on either of these laws do involve elements of chance, sorelations between the measured quantities are probabilistic.(d) Another term which requires some discussion is randomness. A randomaction cannot be predicted and so is due to chance. A random sample is onein which every member of the population has an equal likelihood of appear ing. Just which items appear in the sample is determined completely bychance. If some items are more likely to appear in the sample than others,then the sample is not random.2

Introduction: Probability and Statistics1.2 What does this book contain?We will start with the basics of probability and then cover descriptive statistics. Thenvarious probability distributions will be investigated. The second half of the bookwill be concerned mostly with statistical inference, including relations between twoor more variables, and there will be introductory chapters on design and analysis ofexperiments. Solved problem examples and problems for the reader to solve will beimportant throughout the book. The great majority of the problems are directlyapplied to engineering, involving many different branches of engineering. They showhow statistics and probability can be applied by professional engineers.Some books on probability and statistics use rigorous definitions and many deriva tions. Experience of teaching probability and statistics to engineering students has ledthe writer of this book to the opinion that a rigorous approach is not the best plan.Therefore, this book approaches probability and statistics without great mathematicalrigor. Each new concept is described clearly but briefly in an introductory section. In anumber of cases a new concept can be made more understandable by relating it toprevious topics. Then the focus shifts to examples. The reader is presented with care fully chosen examples to deepen his or her understanding, both of the basic ideas andof how they are used. In a few cases mathematical derivations are presented. This isdone where, in the opinion of the author, the derivations help the reader to understandthe concepts or their limits of usefulness. In some other cases relationships are verifiedby numerical examples. In still others there are no derivations or verifications, but thereader’s confidence is built by comparisons with other relationships or with everydayexperience. The aim of this book is to help develop in the reader’s mind a clear understanding of the ideas of probability and statistics and of the ways in which they areused in practice. The reader must keep the assumptions of each calculation clearly inmind as he or she works through the problems. As in many other areas of engineering,it is essential for the reader to do many problems and to understand them thoroughly.This book includes a number of computer examples and computer exerciseswhich can be done using Microsoft Excel . Computer exercises are included be cause statistical calculations from experimental data usually require many repetitivecalculations. The digital computer is well suited to this situation. Therefore a bookon probability and statistics would be incomplete nowadays if it did not includeexercises to be done using a computer. The use of computers for statistical calcula tions is introduced in sections 3.4 and 4.5.There is a danger, however, that the reader may obtain only an incompleteunderstanding of probability and statistics if the fundamentals are neglected in favorof extensive computer exercises. The reader should certainly perform several of themore basic problems in each section before doing the ones which are marked ascomputer problems. Of course, even the more basic problems can be performed usinga spreadsheet rather than a pocket calculator, and that is often desirable. Even if aspreadsheet is used, some of the simpler problems which do not require repetitive3

Chapter 1calculations should be done first. The computer problems are intended to help thereader apply the fundamental ideas in conjunction with the computer: they are not“black-box” problems for which the computer (really that means the original pro grammer) does the thinking. The strong advice of many generations of engineeringinstructors applies here: always show your work!Microsoft Excel has been chosen as the software to be used with this book for tworeasons. First, Excel is used as a general spreadsheet by many engineers and engi neering students. Thus, many readers of this book will already be familiar with Excel,so very little further time will be required for them to learn to apply Excel to prob ability and statistics. On the other hand, the reader who is not already familiar withExcel will find that the modest investment of time required to become reasonablyadept at Excel will pay dividends in other areas of engineering. Excel is a veryuseful tool.The second reason for choosing to use Excel in this book is that current versionsof Excel include a good number of special functions for probability and statistics.Version 4.0 and later versions give at least fifty functions in the Statistical category,and we will find many of them useful in connection with this book. Some of thesefunctions give probabilities for various situations, while others help to summarizemasses of data, and still others take the place of statistical tables. The reader iswarned, however, that some of these special functions fall in the category of “black box” solutions and so are not useful until the reader understands the fundamentalsthoroughly.Although the various versions of Excel all contain tools for performing calcula tions for probability and statistics, some of the detailed procedures have beenmodified from one version to the next. The detailed procedures in this book aregenerally compatible with Excel 2000. Thus, if a reader is using a different version,some modifications will likely be needed. However, those modifications will notusually be very difficult.Some sections of the book have been label

one hand, it is suitable for practicing engineers in industry who need a better under standing or a practical review of probability and statistics. On the other hand, this book is eminently suitable as a textbook on statistics and probability for engineering students. Areas of practic