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In wireless sensor network, sensory readings are often noisy due to the imprecision of measuring hardware and the disturbance of deployment environment, so it is often inaccurate if we use individual sensor readings to answer queries. In this paper, we consider a useful application of sensor network: maximum average value region query. This query returns the region with the maximum average value among all possible regions in the network, where the region is a fix-sized circle pre-defined by users. Using the average value of a region to answer the query, noises between sensors will be neutralized with each other, which will make the results more reliable. However, because of the huge amount of possible regions in the network, it is costly to process the query exactly. Therefore, we propose a sampling-based algorithm AMAVR to deal with the problem approximately. AMAVR uses a background value to prune the useless regions which cannot be the result. A further optimization strategy is also given to handle the situation that, background value based filter does not work when some individual sensor nodes have higher values than their neighbors. By using both of the two techniques, the scale of the sampling population can be effectively reduced, that is, we cost less energy to get a satisfying result. At last, the conducted simulations demonstrate the energy efficiency of the proposed methods in our paper.