In this dissertation, the problems of computing confidence interval, tolerance interval and prediction interval based on the samples with non-detectable values (i.e., type I censored samples) from normal and related distributions are addressed. Firstly two types of imputation approach have been investigated: one based on the maximum likelihood estimates (MLEs) of the parameters, and the second uses some ad hoc estimates that are particularly suitable for sample sizes that are small or moderately large. Secondly we have investigated the inferential problems concerning the arithmetic mean and the quantiles of a lognormal distribution based on censored samples. Here we have used procedures based on generalized generalized variable approach and modified signed log-likelihood ratio test (MSLRT) statistics. In our investigation we have compared the performance of these two procedures along with that of the signed log-likelihood ratio test (SLRT) statistic for inference concerning the mean and quantiles of a lognormal distribution. Monte Carlo simulation is used to investigate the performance of our procedures. For each of the problems considered, the results are illustrated using practical examples.