Peam and Chen (1998) proposed a generalization of the widely used process capability index (PCI) Cpk to handle processes with asymmetric tolerances. They investigated the sampling distribution and obtained the exact formulae for the expected value and variance of its natural estimator. Recently, Peam and Lin (2000) considered a different estimator under different process condition, and investigated the statistical properties of the new estimator. However, their efforts focused on the small sample properties under the normality assumption. In this paper, we investigate the large sample properties of its natural estimator under the general condition. Based on the limiting distribution of the new estimator, we provide an approximate 100(1 - α)% confidence interval of the considered PCI. The obtained confidence interval provides great benefit to quality engineers on monitoring the process and assessing process performance.