Usually, if the battery OR the AC power dies on mine(ADEMCO), I get a beeping at the control panel and an indication on the LCD that one or the other is out. The AC power missing is not usually a fatal error because it is expected from time to time that you will lose power, which is why there is a battery backup. If you don't have a good battery, the warning is more persistant because if you lose AC power you will have no protection.
When you install a new battery, you need to be sure it is the same voltage and it is hooked up with the proper polarity or it will not work or may even harm your system. You should also be sure the battery is charged before intallation too -- check with a voltmeter.
I'm not sure what you mean by "ringing" but if you are concerned with it calling your monitoring company during these tests, unplug the phone line...