Statistics are impossible with regard to prayer. It's like saying, "How many miles are there in a pound?" Prayer is not a scientific pursuit because there are too many parameters and unknowns to structure a reliable experiment, and therefore statistical analysis and reliability is out of reach. For instance, here are a few things you'd have to know to scientifically assess the effectiveness of prayer:
1. We have to be able to isolate those events on Earth that are actions of God and those that aren't. If we can't create clean categories here, our data may be tainted.
2. We have to be able to guarantee that only certain people (and none others anywhere else in the world) are praying in a certain way for a certain outcome. Any stray prayers unknown to the researchers may skew the data. In addition, we would have to know that absolutely no one in the world was praying for those in the control group. One pray-er, again, may skew the data, and therefore any statistical conclusions. If we can't guarantee exactly who's praying with absolute certainty, then the data may be invalidated.
3. We have to establish objective criteria for what constitutes an answer to prayer and what doesn't. After all, in the Bible God at times uses very normal people and normal circumstances to answer prayer. If we can't define clearly what constitutes an answer to prayer, then the data is invalid. Also, sometimes God answers prayer not in the ways people prayed, but in other ways to answer their prayer by arriving at a different end by a different means, but still what they prayed for. We'd have to be able to define that. And sometimes God answers prayer partially. We have to be able to define that.
We cannot expect reliable and repeatable results suitable for scientific and statistical analysis. Prayer isn't like that. We can't expect to be the ones holding the cards and managing the output. Prayer isn't like that. We can't expect remarkably better results from a scientific and statistical viewpoint. Prayer isn't like that either.
We know that God answers prayer didactically, not empirically. Causation (of any kind) can't be measure empirically without fully isolating variables and replicating results. Revelation ( = being told by God) is the only way we know ANYTHING about what God is like or how God acts. Generally, when we affirm something as an "answer to prayer," this is not on the basis of an absence of physical/biological efficient causes, but on the belief that God works by means of those causes.
There is also the truth that the purpose of prayer is not to motivate God to do something. God does what God will do according to His wisdom, which is not ultimately contingent on anything that anyone else does. He can choose to respond to human input or choose to ignore it, depending on many conditions and complexities. This is a corollary of a divine attribute called Aseity. Christians who know their theology should already affirm this.
So if you were trying to produce a defeater for Christian theology, this isn't one. I would have given the same answer if you'd just asked, "How does prayer work?" As I hear it, your conception of "God answers prayer" is people who pray for things [would] get them at a rate better than random chance would predict (ah, statistics again). You KNOW that this is not how a Christian understands "God answers prayer." So now this is the question you need to ask: What use do you have for a God who will not give you things you ask him for?
If your answer turns out to be "none at all," than nothing I (or anything in Christianity) can say can help you. We do not serve God because we get things from him. God cures our sins and makes us like him, and that has nothing to do with answering our prayers (unless that is what we are praying for, which it should be, and note that these things can't be empirically measured). If the answer is anything else, however, this issue is really a technicality. Why do we pray if not to motivate God to action? Why does God not make his existence self-evident (in this case by answering prayers?) What is the significance of God hearing and acknowledging our prayers if he does not intend to respond? Theology can answer all of these (some more clearly than others), but these discussions are really only apprehensible after divine existence is established; you can't really debate the character and behavior of something that doesn't exist.