The choice of the abscissa-values in linear regression is considered as a decision problem and it is treated in the present paper for the case that this choice is connected with given costs and that the risk is calculable by means of a given quadratic lossfunction. Three optimisation problems are formulated: i) minimal costs with bounded risk - ii) minima! risk with bounded costs - iii) minimal s u n of costs and risk. For these problems the existence of solutions is proved under assumptions concerning the cost function, assumptions being always satisfied in practice. For the first two problems alternative theorems are stated facilitating their treatment essentially.