Fundamentals of Probability and Statistics for Engineers

304 Fundamentals of Probability and Statistics for EngineersandP… 2 > 752:3† ˆ0:95:9.3.2.4 Confidence Interval for a ProportionConsider now the construction of confidence intervals for p in the binomialdistributionIn the above, parameter p represents the proportion in a binomial experiment.Given a sample of size n from population X, we see from Example 9.10 that anunbiased and efficient estimator for p is X. For large n, random variable Xisapproximately normal with mean p and variance p1 p)/n.Defining p…1 p† 1=2U ˆ…X p† ; …9:146†nrandom variable U tends to N(0,1) as n becomes large. In terms of U, we havethe same situation as in Section 9.3.2.1 and Equation (9.129) givesThe substitution of Equation (9.146) into Equation (9.147) gives" #p…1 p† 1=2P u =2 < …X p† < u =2 ˆ 1 : …9:148†nIn order to determine confidence limits for p, we need to solve for p satisfyingthe equation p…1 p† 1=2jX pj u =2 ;nor, equivalentlyp X …k† ˆp k …1 p† 1 k ; k ˆ 0; 1:P… u =2 < U < u =2 †ˆ1 : …9:147†…X p† 2 u2 =2p…1 p†: …9:149†nTLFeBOOK