We consider a classical problem in simulation/statistics - given i.i.d. samples of a rv, the goal is to arrive at a confidence interval (CI) of a pre-specified width varepsilon, and with a coverage guarantee that the mean lies in the CI with probability at least 1-delta for pre-specified deltain(0,1). This problem has been well studied in an asymptotic regime as varepsilon shrinks to zero. The novelty of our analysis is the derivation of the lower bound on the number of samples required by any algorithm to construct a CI of varepsilon -width with the coverage guarantee for fixed varepsilon > 0 and delta, and construction of an algorithm that, under mild assumptions, matches the lower bound. For simplicity, we present our results for rv belonging to a single parameter exponential family, and illustrate its efficacy through a numerical study.