They are correct, your alternator should put out between 13 and 14v when the engine is running. Also, your alternator is internally regulated, meaning there is no other little part that could be causing the problems. That being said, if you really want to be sure, you can take a volt meter to the hot post on the back of the alternator while the engine is running and confirm that there is only 12v there. If this is the case, you need a new alternator.
If, for some incredibly weird reason, you have 13-14v at the alternator, you need to start chasing wires and look for a broken wire or a bad connection.
Also, be forwarned that if you get an arm/sleeve/ponytail/anything caught up in the fan or the belt while the engine is running, you are going in there with it. Just watch your fingers and stay away from moving parts.