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Monitoring quality characteristics of any product or engineering tool requires two techniques. First, Statistical Quality Control (SQC) or Process Control, which is about product inspection at ongoing process level, Second, Acceptance Sampling Plan or Product Control, which decides about the acceptance or rejection of the lot on the basis of sample inspection when the product is ready to explore in the market. At product control, the product under inspection either fails or survive till experiment termination time, and number of defectives observed follows some statistical distribution i.e., Binomial Distribution and total time the product survived is considered lifetime or failure time of the product. The purpose of this thesis is to bring up new sampling distributions, namely Zero Modified Binomial Sampling Distribution and Trinomial Sampling Distribution. Furthermore, our study presents new time censoring schemes, for specialty tests, namely, testing of export quality lot, accelerated tests, and producer or consumer specific tests. The quality characteristics under study is quantile life of the product and the lifetime of the product follows Weibull Distribution.