We introduce a "Statistical Query Sampling" model, in which the goal of an algorithm is to produce an element in a hidden set S \subseteq {0,1}ⁿgn with reasonable probability. The algorithm gains information about S through oracle calls (statistical queries), where the algorithm submits a query function g(.) and receives an approximation to Prx\epsilonS[g(x) = 1] . We show how this model is related to NMR quantum computing, in which only statistical properties of an ensemble of quantum systems can be measured, and in particular to the question of whether one can translate standard quantum algorithms to the NMR setting without putting all of their classical post-processing into the quantum system. Using Fourier analysis techniques developed in the related context of statistical query learning, we prove a number of lower bounds (both informationtheoretic and cryptographic) on the ability of algorithms to produces an x \epsilon S, even when the set S is fairly simple. These lower bounds point out a difficulty in efficiently applying NMR quantum computing to algorithms such as Shor's and Simon's algorithm that involve significant classical post-processing. We also explicitly relate the notion of statistical query sampling to that of statistical query learning.