In finite population sampling, it has long been known that, for small sample sizes, when sampling from a skewed population, the usual frequentist intervals for the population mean cover the true value less often than their stated frequency of coverage. Recently, a non-informative Bayesian approach to some problems in finite population sampling has been developed, which is based on the 'Polya posterior'. For large sample sizes, these methods often closely mimic standard frequentist methods. In this paper, a modification of the 'Polya posterior', which employs the weighted Polya distribution, is shown to give interval estimators with improved coverage properties for problems with skewed populations and small sample sizes. This approach also yields improved tests for hypotheses about the mean of a skewed distribution.