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Given access to independent samples of a distribution A over [n] × [m], we show how to test whether the distributions formed by projecting A to each coordinate are independent, i.e., whether A is ε-close in the L1 norm to the product distribution A1×A2 for some distributions A1 over [n] and A2 over [m]. The sample complexity of our test is O˜(n23/m13/poly(ε-1)), assuming without loss of generality that m≤n. We also give a matching lower bound, up to poly (log n, ε-1) factors. Furthermore, given access to samples of a distribution X over [n], we show how to test if X is ε-close in L1 norm to an explicitly specified distribution Y. Our test uses O˜(n12/poly(ε-1)) samples, which nearly matches the known tight bounds for the case when Y is uniform.