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{sub c}=T{sub c}{sup 0}{l_brace}1+c{sub 1}an{sup 1/3}+c{sub 2}{sup '} ln(an{sup 1/3})+c{sub 2}a{sup 2}n{sup 2/3}+O(a{sup 3}n){r_brace}, where T{sub c}{sup 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 et al., Phys. Rev. Lett. 79, 3549 (1997) in order to understand the difference in the value of the coefficient c{sub 1} between their results and the (apparently) other reliable results in the literature. Our results show that {l_brace}(T{sub c}-T{sub c}{sup 0})/T{sub c}{sup 0}{r_brace}/(an{sup 1/3}) depends strongly on the interaction strength an{sup 1/3} while the previous PIMC results are considerably flatter and smaller than our results. We obtain c{sub 1}=1.32{+-}0.14, in agreement with results from recent Monte Carlo methods of three-dimensional O(2) scalar {phi}{sup 4} field theory and variational perturbation theory.