In this paper, a hop-by-hop Quality-of-Service (QoS) routing mechanism is proposed to maintain the routing performance in networks with imprecise link state information. The proposed mechanism, Normal- PI, applies statistical methods to compute two-sided simultaneous prediction intervals from sampled history of the available bandwidths. A QoS metric - statistical available bandwidth - is then generated from the prediction interval. This metric replaces the instantaneous available bandwidth in the ordinary widest-shortest path routing algorithm (WSR). Simulation results demonstrate that the Normal-PI achieves much less update message overhead compared to WSR without deteriorating the routing performance in terms of packet loss and link utilization.