Believers all around the world claim that their particular god answers petitionary prayers. An answered prayer is a request that is granted while an unanswered one is not, okay? From my experience all that's going on is something called selective observation, where a believer counts the hits and discounts the misses. Scientific studies have shown that these prayers don't get answered any better than luck. So if believers really want to know if God answers prayer then here's what to do:
Keep track, okay, just like the examples in my study sheet. List the specific prayers you pray. Pray that they are answered within a specific time period--why not? Isn't that what you want? Keep in mind that you generally will only pray for things which you expect can happen too. You do not pray that a mountain is uprooted and planted in the sea. So already your prayer requests are limited and already you do not pray for the kinds of things Jesus told you could happen because you live in a scientific era.
Then be brutally honest with the results. No fudging like horoscope readers regularly do. Was the prayer answered exactly as you prayed it, or not? No punting to what believers around the globe do, either. No saying, "well God knows best," or that "he didn't give me what I wanted but what I needed." Then see what happens. Then see how many times your prayers are answered, and of those prayers that are answered how many of them are truly out of the ordinary? I suspect if you really want to know if God answers prayers you'll find this an interesting exercise.
Pray for some miracles while you're at it, like for an amputee to have his limb grow back, or for your God to change a tragic accident the night before so that a carload of kids did not die in that car crash--hey, why not? If God can foreknow your prayers then he could do that, just as you pray when you hear there has been an accident that no one was hurt even though it already happened.