In previous work [1] I have presented an interval estimation procedure applicable to singly censored samples from any univariate population depending only on a scale parameter and location parameter; the procedure applies to several censoring criteria. For any parametric function, the procedure gives an interval estimate based on two confidence statements each at the 1 – 2α level; these estimates were shown to be at least at the (1 – 2α)2level for any size sample and converge, for large samples, to population values. The present work concerns itself specifically with censored samples fromN(μ, σ) and studies the possibility of reducing the conservatism of the procedure. It is shown that, asymptotically, the procedure is at least at the 1 – 2α level. For some special parametric functions, theoretical considerations and fairly extensive computations strongly suggest that the procedure is at least at the 1 1 – 2α level even for small samples. It is conjectured that this property carries over to other functions and other distributions involving only a location and a scale parameter.