  In this paper, we consider the nonasymptotic sequential estimation of means
of random variables bounded in between zero and one. We have rigorously
demonstrated that, in order to guarantee prescribed relative precision and
confidence level, it suffices to continue sampling until the sample sum is no
less than a certain bound and then take the average of samples as an estimate
for the mean of the bounded random variable. We have developed an explicit
formula and a bisection search method for the determination of such bound of
sample sum, without any knowledge of the bounded variable. Moreover, we have
derived bounds for the distribution of sample size. In the special case of
Bernoulli random variables, we have established analytical and numerical
methods to further reduce the bound of sample sum and thus improve the
efficiency of sampling. Furthermore, the fallacy of existing results are
detected and analyzed.
