A summary is given of the current state of knowledge concerning the analysis of data arising from Weibull populations. Five different parameterizations which have been used by various authors are given, and the relation of the various sets of parameters to each other and to the mean and the variance is explored. It is shown that the standardized cumulants are functions only of the shape parameter. An auxiliary table is given of the mean and the variance of a Weibull population with location parameter 0 and scale parameter 1 and of the standardized cumulants gamma sub k (k = 3,4 ...,8), all for shape parameter m = 1.1(0.1)10.0. The theory is developed for tests of hypotheses concerning the mean and the variance of a Weibull population, based on the Weibull-Z, Weibull-T, and Weibull-V statistics, which are analogues of the normal-z, Student-t, and (chi square)/(degrees of freedom) statistics, respectively, the difference being that the underlying distribution is Weibull rather than normal. Percentage points of the Weibull-Z statistic are worked out by use of the Cornish-Fisher expansion technique and similar results for the Weibull-V and Weibull-T statistics are obtained by means of a Monte Carlo simulation. The percentage points are tabulated for all combinations of the above values of m; sample sizes n = 2(1)40, 48, 60, 80, 120, 240, infinity; and cumulative probabilities P = 0.005, 0.01, 0.025, 0.05, 0.1, 0.25, 0.75, 0.9, 0. 95, 0.975, 0.99, 0.995. Examples illustrating the use of the tables are given.