My civic without any rhymne or reason occasionally won't start

The battery was replaced on advice from an AA patrolman. this appeared to cure the fault but this lasted only a month then although the car was working perfectly well when parked for the night,in the morning without any warning it might not start. I have tried disconnecting the battery at night and reconnecting in the morning,so far the car has started O.K When I got an electrics test done the altenator was putting out around 12.8 amps but during this test the engine started hunting occasionally after the test and the engine was switched off there was no output from the battery. I don't want to go to the expense of a new altenator without concrete proof that is the fault I really hope you can help with this