The problem of finding a confidence interval of preassigned length and of more than a given confidence coefficient for the unknown mean of a normal distribution with unknown variance is insoluble if the sample size used is fixed before sampling starts. In this paper two-sample plans, with the size of the second sample depending upon the observations in the first sample (as in [1]), are discussed. Consideration is limited to those schemes which increase the center of the final confidence interval by $k$ if each observation is increased by $k$, and for which the size of the second sample is a function only of the differences among the observations in the first sample. Then it is shown that the mean of all the observations taken should be used as the center of the final confidence interval. Those schemes which make the size of the second sample a nondecreasing function of the sample variance of the first sample are shown to have certain desirable properties with respect to the distribution of the number of observations required to come to a decision.