Sequential decisions procedures for the examination of the unknown mean of a normal distribution are given. The indifference regions of this procedures are easily constructable triangular areas, which will be chosen so, that the procedures observe the given probabilities of errors or risks respectively. Computational investigations indicate, that the maximum expected value of hte sample size required by the Wald's method. Further on we investighate the quality of the procedures in the Bayes case, when the unknown parameter has certain general apriori distributions.