This paper studies the performance of multistage sequential sampling procedures in chronological order, starting from Stein’s two-stage procedure, the one-by-one purely sequential procedure, Hall’s three-stage procedure, and the accelerated sequential procedure for estimating the mean of the normal distribution under a moderate sample size using Monte Carlo simulation. We also introduce and discuss the performance of a new sequential sampling procedure called the progressive procedure that starts with a bulk stage and ends by one-by-one purely sampling under moderate and large sample sizes (asymptotic) based on Monte Carlo simulation. The simulation results show that the new procedure competes with other procedures and attains all targeted asymptotic characteristics except the exact consistency property.