In this paper we investigate transmission scheduling by Medium Access Control (MAC) for energy-efficient detection using Wireless Sensor Networks (WSN). We consider the binary hypothesis testing problem. The decision is made by an access point and is based on received data from sensors that transmit through a fading channel. We study the significance of exploiting both Channel-State Information (CSI) and Likelihood-Ratio Information (LRI) to design an adequate MAC protocol that minimizes the total transmission energy required for optimal detection. We formulate the access problem as a history-dependent decision process. The optimal solution is mathematically intractable and suffers from exponential complexity as a function of model size. Hence, we propose an approximate solution using the Markov property to reduce complexity and make the problem mathematically tractable. We designed the LRI and CSI Based Access (LCBA) protocol based on this solution. The LCBA protocol trades off between LRI and CSI to reduce the total transmission energy. Simulation results show a significant performance gain of LCBA over existing approaches.