This paper reports a simulation-based exploration into the computation of point and interval estimates for data arising from sequential sampling of populations with a variety of underlying distributions. We conclude that the coverage probability of the standard frequentist confidence interval estimates is overstated. The effect of non-normal behavior in the underlying population upon the properties of the interval estimate varies, depending upon the severity and type of anomaly (skewness, kurtosis, etc). Sequential sampling of certain nonsymmetric distributions performed very poorly compared with simple random sampling. We assess other interval estimates that do not overstate the coverage probability if the underlying population is normal (Gaussian).