The sequential-probability-ratio-test (SPRT) control chart is well-known for its outstanding detection ability towards a wide range of process mean shifts. In the existing literature, the SPRT chart has been regularly designed from the standpoint that the process parameters are known. Nonetheless, industrial practitioners reveal that these process parameters are hardly known. They often estimate these process parameters using a modest number of Phase-I samples. This in turn leads to different outcomes obtained across practitioners. In this paper, we present detailed insights about the distribution of the conditional ATS when the process parameters are estimated. Also, some statistical properties of the ATS through Monte Carlo simulation are presented in this paper. Particularly, the standard deviation of the ATS is examined in this paper to evaluate the practitioner-to-practitioner variability. Results show that a large Phase-I sample, and a large desired maximum mean shift in optimization design, are required to reduce volatilities in the average and standard deviation of the ATS values. Finally, we provide recommendations for the Phase-I sample sizes tailored to specific objectives of the SPRT chart when the process parameters are estimated.