We consider independent random variables (r.v.'s) with a common mean $μ$ that either satisfy Lindeberg's condition, or are symmetric around $μ$. Present forms of existing functional central limit theorems (FCLT's) for Studentized partial sums of such r.v.'s on $D[0,1]$ are seen to be of some use for constructing asymptotic confidence intervals, or what we call functional asymptotic confidence intervals (FACI's), for $μ$. In this paper we establish completely data-based versions of these FCLT's and thus extend their applicability in this regard. Two special examples of new FACI's for $μ$ are presented.