  Using a finite-temperature Path Integral Monte Carlo simulation (PIMC) method
and finite-size scaling, we have investigated the interaction-induced shift of
the phase transition temperature for Bose-Einstein condensation of homogeneous
weakly interacting Bose gases in three dimensions, which is given by a proposed
analytical expression $T_{c} = T_{c}^{0}\{1 +
c_{1}an^{1/3}+[c'_{2}\ln(an^{1/3})+c''_{2}]a^{2}n^{2/3} +O(a^{3}n)\}$, where
$T_{c}^{0}$ is the critical temperature for an ideal gas, $a$ is the s-wave
scattering length, and $n$ is the number density. We have used smaller number
densities and more time slices than in the previous PIMC simulations [Gruter
{\it et al.}, Phys. Rev. Lett. {\bf 79}, 3549 (1997)] in order to understand
the difference in the value of the coefficient $c_{1}$ between their results
and the (apparently) other reliable results in the literature. Our results show
that $\{(T_{c}-T_{c}^{0})/T_{c}^{0}\}/(an^{1/3})$ depends strongly on the
interaction strength $an^{1/3}$ while the previous PIMC results are
considerably flatter and smaller than our results. We obtain $c_{1}$ = 1.32
$\pm$ 0.14, in agreement with results from recent Monte Carlo methods of
three-dimensional O(2) scalar $\phi^{4}$ field theory and variational
perturbation theory.
