2006 HCSDB Adult Sampling Report - Tricare

7HFKQLFDO%DFNJURXQGIRUWKH$OJRULWKPTo attain the required half length HL for confidence intervals, the required sample size n wasobtained while incorporating finite population correction factors that recognized that the geographicareas and beneficiary and enrollment groups had variable population sizes.For a simple random sample (SRS) of size n from a finite population of size N, the variance of asample proportion p is:(G.1)VSRSP (1- P) ⎛ N - n ⎞(p) = ⎜ ⎟n ⎝ N - 1⎠where P denotes the population proportion. Because the expected sample sizes for all strata forthe 2006 HCSDB survey are sufficiently large, the standard formula (IV.1) in Chapter IV can beused in constructing the confidence interval of P. Let B denote the required half-length interval forP. Using formula (G.1) for the simple random sample variance of p, the precision requirement Bcan represented by the following equation:B = z(G.2) 1- α /2P (1 - P) ⎛ N - n ⎞⎜ ⎟n ⎝ N - 1⎠Consequently, the sample size to attain the precision requirement B can be determined by solvingequation G.2 with respect to n as follows:(G.3)2z1- α /2 [P(1- P)]2n = B21 ⎛ z1- α /2 [P(1- P)] ⎞1+2N⎜⎟⎝ B ⎠This formula was used as the first step in determining initial sample sizes for all strata in the 2006HCSDB.Note from formula (G.3), sample sizes vary according to values of the proportion P. As the valueof P becomes closer to 0.5, n becomes larger. Because proportions of interest for this surveycould have values ranging from zero to one, the resulting sample sizes lie within a wide range ofvalues with the largest value associated with P=0.5. For sample size determination, we used aproportion value of P=0.5, which ensures that the sample size will be large enough to meet orexceed the predetermined precision requirement for all proportions to be estimated.Since the sample size is being defined to construct a 95 percent interval for P = 0.5 with a halflengthinterval less than or equal to B, z /2 - 1 α can be replaced with z.975 which is 1.96. Formula(G.3) can then be specified as the following:G-1