Will apple cover this with applecare on my iphone 5?

I bought an iPhone 5 off Craigslist and it uses Sprint and has no Sim Card. The only problem is that the phone shuts off ***SOMETIMES*** not always at low battery. Sometimes it shuts off at 30, other times at 20. I have applecare until November. When my iphone 4 screen wouldn't turn on, the apple store gave me a new one for free. Will they do the same for my iphone 5?

If it was never jailbroke and has no water damage, they should replace it with a referb

Back up your phone in itunes then restore device as new. Don't put back up on and use the phone to see if it works ok without your stuff on it. If it still shuts off then take it to the apple store.

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