Joint prediction intervals (based upon the original fitted model) for K future responses at each of K separate settings of the independent variable have been treated by Lieberman (1961). When K is unknown and possibly arbitrarily large, these results do not apply. A solution to the problem of arbitrary K is given in terms of tolerance intervals on the distributions of future observations, the intervals being (probabilistically) simultaneous in each possible value of the independent variable. Four alternative techniques are proposed and compared for their applicability in different situations. The first is the simultaneous extension of the Wallis (1951) technique. The other three are based on Scheffé simultaneous confidence principles. One gives intervals for a fixed central proportion P of the distribution which are simultaneous in all values of the independent variable; the other two give intervals simultaneous in the independent variable and different central proportions P. A numerical example is analysed, and some remarks are made on the applicability of the Scheffé techniques to the detection of outliers.