Fundamentals of Probability and Statistics for Engineers

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