Fundamentals of Probability and Statistics for Engineers

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Parameter Estimation 297f U (u)1– αα /2 α /2−u α /2 0 u α /2uFigure 9. 6 [100(1)]% confidence limits for UP… u =2 formation given by Equation (9.123), we have the generalresultP Xu =2n 1=2< m < X ‡ u =2n 1=2ˆ 1 : …9:130†This result can also be used to estimate means of nonnormal populations withknown variances if the sample size is large enough to justify use of the centrallimit theorem.It is noteworthy that, in this case, the position of the interval is a function ofX and therefore is a function of the sample. The width of the interval, incontrast, is a function only of sample size n, being inversely proportional to n 1/2 .The [100(1 )] % confidence interval for m given in Equation (9.130) alsoprovides an estimate of the accuracy of our point estimator X for m. As we seefrom Figure 9.7, the true mean m lies within the indicated interval with[100(1 )] % confidence. Since X is at the center of the interval, the distanceσX – uX —σα / 2m X + un 1/2 α / 2dn 1/2Figure 9.7Error in point estimator Xfor mTLFeBOOK
298 Fundamentals of Probability and Statistics for Engineersbetween X and m can be at most equal to one-half of the interval width. Wethus have the result given in Theorem 9.6.)]% con-Theorem 9.6: let X be an estimator for m. Then, with [100(1fidence, the error of using this estimator for m is less thanu /2n 1/2Ex ample 9. 18. Problem: let population X be normally distributed withknown variance 2 .IfX is used as an estimator for mean m, determine thesample size n needed so that the estimation error will be less than a specifiedamount " with [100(1 )] % confidence.Answer: using the theorem given above, the minimum sample size n mustsatisfy" ˆ u=2n : 1=2Hence, the solution for n isn ˆu=2 2:…9:131†"9.3.2.2 Confidence Interval for m in N(m, s 2 ) with Unknown s2The difference between this problem and the preceding one is that, since is notknown, we can no longer use 1U ˆ…X m†n 1=2as the random variable for confidence limit calculations regarding mean m. Letus then use sample variance S 2 as an unbiased estimator for2 and consider therandom variable S 1Y ˆ…X m† : …9:132†n 1=2The random variable Y is now a function of random variables X and S. Inorder to determine its distribution, we first state Theorem 9.7.Theo re m 9 . Student 7 : ’s t-dist ribut ion. Consider a random variable T defined byT ˆ UV 1=2: …9:133†nTLFeBOOK
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ContentsPREFACExiii1 INTRODUCTION 1
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Contentsix6.1.2 Geometric Distribut
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ContentsxiAPPENDIX A: TABLES 365A.1
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PrefaceThis book was written for an
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1IntroductionAt present, almost all
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Introduction 3may be rejected at th
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Part AProbability and Random Variab
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390 Subject IndexExtreme-value dist
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