Initial sample size problem in the sequential test for the mean of a normal distribution

The two-stage sequential test, suggested by Baker [2] for testing hypotheses $H_0:\mu=\mu_0$ and $H_1:\mu=\mu_1$ of $N(\mu,\sigma^2)$ with the unknown $\sigma^2$ would not be amenable for applications unles some cluses on the choice of the first-stage sample size are available. The study in this paper is intended to shed some light on the size of the first-stage sample. An approximate method is used to estimate an optimal initial sample size that minimizes the average sample number. In brief, the optimal size is a strictly monotone decreasing function of the quantity $(\mu_1-\mu_0)/\sigma$. Empirical and simulation results are used to ascertain the negligible effect of possible errors due to approximations and assumptions used.