My thoughts after cut (briefly, still gotta cook).
I confess I'm not sure if this is worth doing. On the one hand, I find the argument that "it's stupid to waste tax payer money on something we know is bunk" enormously unconvincing. Science and medicine generally is filled with examples of things thought to be bunk or superstition that subsequently had basis in fact, or where science fact later turned out to be bunk. Some of the earliest FDA cases involved prosecuting people for making health claims about vitamin supplements, because science clearly showed that vitamin supplements had no impact and the claim that women had different vitamin or nutritional needs was garbage. Meditation and biofeedback were dismissed as Eastern nonesense because how could just thinking about things produce physical effects. Etc., etc.
OTOH, the impact of prayer and distant prayer strikes me as way too difficult to measure and quantify. It also runs into the problem of what if there is a God and there is only one correct religion, as opposed to some underlying non-religious physics principle at work. For example, if I am right in my beliefs that there is one true God and he hates idol worship, then a test which includes Wiccan healers may fail becuase God decides he doesn't want them to succeed. Or the object of the prayer may be a sinner and unworthy. Or perhaps the object of the prayer is worthy, but God has determined this person's fate.
And with that, good shabbos.