The Equivalence of Neyman Optimum Allocation for Sampling and Equal Proportions for Apportioning the U. S. House of Representatives

• Activity Number: 582
• Type: Contributed
• Date/Time: Wednesday, August 1, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Section on Survey Research Methods
• Author(s): Tommy Wright*+
• Companies: U.S. Census Bureau
• Keywords: Apportionment ; Equal Proportions ; Neyman Allocation ; Stratification ; Probability Sampling

Abstract

Tommy Wright U. S. Bureau of the Census

We present a surprising though obvious result which seems to have been unnoticed until now. In particular, we demonstrate the equivalence of two well-known problems - the optimal allocation of the fixed overall sample size n among L strata under stratified random sampling(strs) and the optimal allocation of the H=435 seats among the 50 states for apportionment of the U. S. House of Representatives following each decennial census. In spite of the strong similarity manifest in the statements of the two problems, they have not been linked and they have well-known but different solutions; one solution is not explicitly exact (Neyman allocation), and the other (equal proportions) is exact. We give explicit exact solutions for both, note that the solutions are equivalent, and note that both problems are special cases of a general problem. The result is significant for strs, explicitly showing how to minimize sampling error when estimatng a total while keeping the final sample size fixed at n; this is usually not the case in practice with Neyman allocation where the resulting final overall sample size might be as large as n + L after rounding.