This paper proposes a stochastic model predictive control (SMPC) algorithm for linear discrete-time systems with bounded additive noise and incomplete state information. Firstly, assuming that the first-order and second-order moments of the noise are known. The probability constraints imposed on the state and input are reformulated in deterministic forms by using Chebyshev-Cantelli inequality. Then, an observer-based output feedback controller is designed under the framework of efficient model predictive control (EMPC). Moreover, additional perturbations are introduced and the method of offline computation, online synthesis is adopted to maximize the initial feasible set and calculate the control law. Finally, we provide an average asymptotic performance bound and proof the recursive feasibility of the proposed algorithm. Simulation results are presented at the end to demonstrate the effectiveness of the algorithm.