There are special situations where specification limits on a process are implemented externally, and the product is typically reworked or scrapped if its performance does not fall in the range. As such, the actual distribution after inspection is truncated. Despite the practical importance of the role of a truncated distribution, there has been little work on the theoretical foundation of standardization, inference theory, and convolution. The objective of this research is three-fold. First, we derive a standard truncated normal distribution and develop its cumulative probability table by standardizing a truncated normal distribution as a set of guidelines for engineers and scientists. We believe that the proposed standard truncated normal distribution by standardizing a truncated normal distribution makes more sense than the traditionally-known truncated standard normal distribution by truncating a standard normal distribution. Second, we develop the new one-sided and two-sided z-test and t-test procedures under such special situations, including their associated test statistics, confidence intervals, and P-values, using appropriate truncated statistics. We then provide the mathematical justifications that the Central Limit Theorem works quite well for a large sample size, given samples taken from a truncated normal distribution. The proposed hypothesis testing procedures have a wide range of application areas such as statistical process control, process capability analysis, design of experiments, life testing, and reliability engineering. Finally, the convolutions of the combinations of truncated normal and truncated skew normal random variables on double and triple truncations are developed. The proposed convolution framework has not been fully explored in the literature despite practical importance in engineering areas. It is believed that the particular research task on convolution will help obtain a better understanding of integrated effects of multistage production processes, statistical tolerance analysis and gap analysis in engineering design, ultimately leading to process and quality improvement. We also believe that overall the results from this entire research work may have the potential to impact a wide range of many other engineering and science problems
