Testing means from sampling populations with undefined labels

We consider the problem of testing means from samples of two populations for which the labels are not defined with certainty. We show that this problem is connected to another one that is testing expected values of components of mixture-models from two data samples. The underlying mixture-model is associated with known varying mixing-weights. We provide a testing procedure that performs well. Then we point out the loss of performance of our method due to the mixing-effect by comparing its numerical performances to the Welch's t-test on means which would have been done if true labels were available.