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Statistical hypothesis testing is useful for controlling and improving processes, products, and services. This most fundamental, yet powerful, continuous improvement tool has a wide range of applications in quality and reliability engineering. Some application areas include statistical process control, process capability analysis, design of experiments, life testing, and reliability analysis. It is well-known that most parametric hypothesis tests on a population mean, such as z-test and t-test, require a random sample from the population under study. However, there are special situations in engineering, where the specification limits, such as the lower and upper specification limits, on the process are implemented externally, and the product is typically reworked or scrapped if the performance of a product does not fall in the range. As such, a random sample needs to be taken from a truncated distribution; however, there has been little work on the theoretical foundation of statistical hypothesis procedures under this special situation. The objective of this paper is twofold. First, we provide the mathematical justifications that the central limit theorem works quite well for a large sample size, given samples taken from a truncated distribution. We also verify this finding using simulation. Second, we then develop the new one-sided and two-sided z-test and t-test procedures, including their test statistics, confidence intervals, and P-values, using appropriate truncated statistics.

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