"Consider an expanded version of the placebo affect."
Dosen't work, unless you have determined that prayer ( like a placebo ) has no effect.
You don't even need a test group for that - simply tell one group of patients that they are being prayed for by hundreds of people. If it works as a placebo, then they should do better.
"Prayer matters in the same way that calling my parents matter"
If you are in trouble and you call your parents to "deliver me from evil ( send me money )" you have some expectation that you MAY be "delivered", although you may not.
If you DON'T call them, odds are they will not spontaneously decide to send you a check for no reason.
My comment had to do with some eastern & "new age" beliefs, instead of current common western beliefs. Think in terms of reports by twins "knowing" when harm has come to their other twin, something nearly impossible to reliably document.
If you are in trouble and you call your parents to "deliver me from evil ( send me money )" you have some expectation that you MAY be "delivered", although you may not.
If you are in trouble and you call your parents to "deliver me from evil ( send me money )" you have some expectation that your parent MAY be tell you to get a job or to cut back on expenses. It may not be the kind of relief you wanted, but if you need that kind of attitude adjustment, sending immediate relief could be harmful to you.
If you DON'T call them, odds are they will not spontaneously decide to send you a check for no reason.
Is the only measure of your relationships material?