The power functions of the Studentized range test and the F-test for comparing the means of normal populations in the one-way fixed effects analysis of variance model is investigated. The least favourable configurations of population means under certain range restrictions of the population means are given. These results are important in the calculation of the sample sizes required to guarantee power level under the restrictions on the range of the population means. The optimal test procedures which maximise the least favourable power under the range restrictions of the population means are found for the known variance case. Then the relations between these optimal tests and the range type tests, sum of square tests are pointed out. The problem of comparing several treatment means with a control mean is also considered. Finally, these results are applied to the binary data, by using the arcsin-root transformation and the asymptotic distribution, for both comparison between k Bernoulli responses and comparison between k Bernoulli responses and a control Bernoulli response.