The properties of the usual one-sample T-statistic under nonnormal universes are investigated using Edgeworth expansions, and the findings reinforce the observations made by many authors in the past;Sufficient conditions for combining two independent unbiased estimators of a common mean in order to obtain a uniformly better (in variance sense) unbiased estimator is given. An upper bound for the inefficiency of such an estimator is also presented using a Kantorovich inequality;Heteroskedasticity-robust test procedures for the one-sample and two-sample problems are developed. An unbiased estimator of the variance of the ordinary least squares estimator of the slope parameter of a heteroskedastic simple linear regression model without intercept is given and performance of this estimator is assessed using Monte-Carlo simulations.