A feedback control framework is developed for stabilizing discrete-time switched linear stochastic dynamical systems for the case where the mode of the switched system is not observable at all time instants. We propose a feedback control law that depends only on the periodically obtained imprecise mode information. Specifically, the modes of the switched system are assumed to be divided into a number of groups, and the periodically available mode information indicates only the group that contains the active mode. We obtain sufficient conditions for second-moment asymptotic stability of the closed-loop system under our proposed control law. We demonstrate the efficacy of our approach through an illustrative numerical example.