2 00 8 On Estimation of Finite Population Proportion ∗

In this paper, we study the classical problem of estimating the proportion of a finite population. First, we consider a fixed sample size method and derive an explicit sample size formula which ensures a mixed criterion of absolute and relative errors. Second, we consider an inverse sampling scheme such that the sampling is continue until the number of units having a certain attribute reaches a threshold value or the whole population is examined. We have established a simple method to determine the threshold so that a prescribed relative precision is guaranteed. 1 Fixed Sample Size Method The estimation of the proportion of a finite population is a basic and very important problem in probability and statistics [4, 6]. Such problem finds applications spanning many areas of sciences and engineering. The problem is formulated as follows. Consider a finite population of N units, among which there are M units having a certain attribute. The objective is to estimate the proportion p = M N based on sampling without replacement. One popular method of sampling is to draw n units without replacement from the population and count the number, k, of units having the attribute. Then, the estimate of the proportion is taken as p̂ = k n . In this process, the sample size n is fixed. Clearly, the random variable k possesses a hypergeometric distribution. The reliability of the estimator p̂ = k n depends on n. For error control purpose, we are interested in a crucial question as follows: For prescribed margin of absolute error εa ∈ (0, 1), margin of relative error εr ∈ (0, 1), and confidence parameter δ ∈ (0, 1), how large the sample size n should be to guarantee Pr { |p̂ − p| < εa or ∣∣∣∣ p̂ − p p ∣∣∣∣ < εr } > 1− δ? (1) ∗The author is currently with Department of Electrical Engineering, Louisiana State University at Baton Rouge, LA 70803, USA, and Department of Electrical Engineering, Southern University and A&M College, Baton Rouge, LA 70813, USA; Email: [email protected]