Abstract: In scientific applications, interest usually focuses on the ``superpopulation'' parameters of a stochastic model hypothesized to underlie the generation of the values in a finite population, rather than finite-population parameters themselves. Variance formulas for sampled data that incorporate finite-population correction factors are not appropriate for these applications. For simple random sampling, it is common practice to ignore these correction factors in variance estimation; this yields correct superpopulation inference under a simple superpopulation model. This is shown to hold true for two-stage simple random sampling of clusters, but not for stratified sampling or probability-proportional-to-size sampling. Asymptotically unbiased variance estimators are provided for these latter two types of sampling that are appropriate for superpopulation inference under a general superpopulation model. An application is given using data from the 1987 National Health Interview Survey which shows that the difference between classical repeated-sampling variance estimators and a superpopulation variance estimator can be quite large.
Key words and phrases: Cluster sampling, finite-population correction factors, probability-proportional-to-size sampling, stratified sampling.