In this dissertation, test statistics for determining whether the k $(k\geq2)$ Weibull location parameters are equal are given both for censored and complete samples. For the case when the shape parameter, $\beta,$ of the Weibull distribution is equal to unity, the exponential case, simpler test statistics are provided that are functions of the means and first order statistics The asymptotic distribution of the test statistic is derived for two complete samples from exponential populations. The power of the test of equality of two location parameters for complete samples from exponential populations is derived using a new approach. Ten thousand exponential samples, with the desired mean and location parameter are simulated using the International Mathematical and Statistical Libraries software (Visual Numerics, Inc., Houston, 1991) and the number of times the null hypothesis (of equality of two location parameters) is rejected is then recorded. Simulated power is the proportion of the samples for which the null hypothesis is rejected Generally, there is good agreement between calculated and simulated power. Power depends on the ratio of hazards, sample size, the magnitude of the difference between location parameters and the level of significance. Fixing any three of the preceding parameters and increasing the fourth results in increased power