I couldn't prove it conclusively, but, particularly when I have been praying for other people and then I am indirectly used as part of the answer.
For example: There was one time I was praying for a family in the church I was attending (they were involved in work in the church and relied on donations). I sensed (consider it like a thought) that God was saying to me that I needed to give them some money. When I asked, "How much", the response was, "Enough for a good holiday". So I came up with a figure of what I thought a family of four would need for a good holiday.
When I went to church, I was a bit nervous about telling them this and looking a bit of an idiot, so didn't speak to them at the start of the service. At the end, I decided I had to go for it and spoke to them. I was then told that they were due to go on holiday, but their car had broken down. They had spent all of their spending money on getting it repaired, so what I gave them was virtually identical to what they had spent on the car. Now, I suppose it's possible I have some hidden skills which allow this to happen now and again, but for me, prayer is a better explanation!
I don't know whether this is evidence for the power of prayer, but it certainly makes your god look extremely parochial. I know a young lady, who as a child learned that her single mother was dying of cancer. Day after day, for weeks, she was on her knees praying that her mummy get better. Guess what - her mummy died. Little girl hadn't got the correct Christian belief, or wasn't saying her prayers in the right way? Or something?
All these arguments about 'God answering prayers' are completely worthless, since other explanations are available, and instances such as I've related above abound, and make such a God appear a monster who scatters his favours when he feels like it ( and doesn't do so when he also feels like it).