A method is presented for estimating population from sample proportions that produces margins of error narrower for any specific sample size or that requires a sample size smaller for any specific margin of error than do previously existing methods applied to the same data. This method applies an unbiased estimator of the squared correlation between population and sample proportions to determine point and interval estimates of population proportions in a regression context involving simple random sampling with replacement. In virtually all reasonable applications, assuming a Dirichlet prior distribution, the margin of error produced by this method for a population proportion is shown to be 1.96 times the posterior standard deviation of the proportion.