It is critical to account for uncertainty in the design of transportation networks. Various models assuming both user-optimal and system-optimal behavior (as a computationally viable proxy for the more realistic useroptimal design problem) have been proposed. Most of these models do not provide any form of probabilistic guarantee for the obtained capacity expansion decisions. However, for system reliability, it is often useful to know how likely it is that the total system travel time would deviate from a certain value if the prescribed solutions from a specific model are implemented. A new mean-variance type of system-optimal network design model with probabilistic guarantees on systemwide travel time is proposed. The proposed model has several unique features. First, uncertainty in the link performance function is considered. This uncertainty is a result of capacity uncertainty as well as fundamental uncertainty about the functional form of the link performance function itself. Second, instead of imposing an explicit chance constraint-which in general would lead to nonconvexity-probabilistic guarantees on the obtained system travel time are obtained implicitly. More specific, the model yields a one-sided confidence interval for the total systemwide travel time that has an a priori specified confidence level. Finally, it is not necessary to specify an explicit probability distribution to model the uncertainty. Instead, the proposed model is distribution free in that any symmetric probability distribution suffices. Numerical results are presented and discussed.