X1,X2, …, are independent and identically distributed, the common distribution known except for the value of a parameter θ. The problem is to find a test ofH–1:θ = θ–1againstH1.θ=θ1, with pre-assigned error probabilities α, β, and which minimizes the maximum expected sample size (the maximum taken with respect to θ). For the cases where α = β and θ is the mean of a normal distribution, and where α = β and θ is a binomial probability and θ–1+ θ1= 1, a computational scheme for finding such a test is described.