Yes. "Does the act of prayer have beneficial psychological effects?" and "does God exist?" are two different questions, and I do not presume any dependence of

Yes. "Does the act of prayer have beneficial psychological effects?" and "does God exist?" are two different questions, and I do not presume any dependence of the latter on the former. Besides, I don't see how being dependent upon the results of an experiment has any bearing on the meaning of the question. The data collected in the experiment may be empirically valid, but the connection between the data and the question is inherently rational, not empirical, and you must already have a set of presumptive axioms in order to connect the concrete data to the abstract question.