If when you have the old battery removed and the laptop still works ok then yes it is the battery. Googling laptop battery life will get you lots of conflicting info on the subject. Generally the battery warranty is 1 year. After that you are on your own. The life is dependent upon how the battery is recharged (fully drain before recharging) though for new batteries that is not so much a problem, in theory.
Got a battery depot nearby you can get the battery tested? But I would just go ahead and replace it.