Well taking a second look at that prayer research it seems 50/50 so that was a bad idea on me. It really is a impossible thing for anyone to really prove, it seems to me it has to do with our spirituallity, and I really think its in all of us some people call it something different but we all posses it, I can give a description I will if someone wants me too, but I'll wait on that so I don't get ambiguous with it like I've done a few times and I apologize for that. But I really am a logical person whether you believe me or not I am and even as a kid it was before I was ten I was young, but I became terrified of heaven and eternity and started to talk about as well. That was the first of many, howerever now that same thought is no longer a struggle. They way I verify my own experiences with God is when I think I have an answer from him if I have any question whether it is or is not, any question at all later afterward, any relating to it at all then I know it's not his answer. Only when I can have no question at all on it at any point in time. I can add a little more if you'd like but I'm short on time at the moment.
From the sounds of things, the results of prayer is indistinguishable from the results of non-prayers.
You make prayers, stuff happens, and you assume that god had a hand in it. You see god's hand everywhere. That about sums it up, right?
However, what I see are stuff that had some chances of happening anyway. If someone loses his keys (you didn't label any of my examples as bad prayers, so I'm assuming all of them are fair game), there is usually some chance of him finding them. People usually have some chance of finding a job, people usually have some chance of recovering from illness, etc etc. In such cases, I see it as very unlikely that god, or any sort of supernatural beings had involvment, since there is nothing to suggust such a thing. It seems far, far more likely that prayer is little more than a form of magical thinking. It might make you feel good, but that's about it.
This is why we ask the amputee question (and other prayers that god doesn't answer). For humans, the odds of them regrowing a limb is virtually nonexistent. For this prayer to be answered, it would require that god does something. It would not be up to "something that could've happened anyway". If we saw such prayers happening, it would get us talking. Very fast. I don't know if it would result in "instant conversion", but it would get us wondering what's going on, instead of wondering why you think magic is happening.
So do you have imprearical proof that no man in the history of the world has never had a limb regenerated if you will, by God. There is no way either of us could do such a thing since man predates written history.
So your saying this one question hold the vast majority of the reason for disbelief?