The paper proposes a novel design of simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) in a wireless powered Internet of Things (IoT) network, where two sensor node groups (SNGs) harvest energy from a power station (PS) and transmit their message to an access point (AP) with the harvested energy. The STAR-RIS, which is deployed in the middle of the SNGs and adopts the time splitting (TS) working mode, can help the energy transfer in the wireless energy transfer (WET) phase and the information transfer in the wireless information transfer (WIT) phase. The paper aims to maximize the sum throughput from the two SNGs to the AP by jointly designing the phase shifts of the STAR-RIS and the working time allocated to the two SNGs in the WET and WIT phases, respectively. To solve the formulated non-convex optimization problem, we propose a low-complexity algorithm where we first derive the optimal phase shifts of the STAR-RIS in the WIT phase. Then, we adopt the Lagrange dual method to simplify the optimization problem and optimize the phase shifts of the STAR-RIS in the WET phase by the Majorization-Minimization (MM) algorithm and the complex circle manifold (CCM) algorithm. Next, a two-layer iterative algorithm is used to obtain the optimal values of time allocated to the two SNGs. Finally, we evaluate the improvement of the proposed scheme by the simulation results compared with other benchmark schemes..