Prayer puzzles me. It always has. I don't think I will ever not be puzzled about prayer - despite people's valiant attempts to explain prayer's innate impotence or interpret my lack of faith in prayer as a bruising consequence of my JW background. Ha! I was cynical about prayer even as a door-knocking JW. It has always been a chink in my armor. Thank God!
There have actually been some (quasi) scientific studies on the effects of prayer in medical outcomes.
I'm on the phone at the moment but if you google 'studies on intercessory prayer' that should give you enough to get you banging your head on the PC screen.