Do I need to connect the ground wire?

Herein, what do you do if the ground wire has no ground?

If the box is not grounded, you have two choices:

1. Run a ground to the box. This would mean running a new wire back to the panel (could be bare copper, but if you're doing the work may be better just to run a new wire - 14/3 would be the best), which could mean opening up drywall.
2. Wire the fan without ground.

One may also ask, how do you run a ground wire? Attach the grounding wire to the grounding terminal. If the grounding wire has become loose or disconnected, loop the grounding wire over the green terminal screw and tighten with a Philips or flathead screwdriver. Make a loop at the end of the copper wire with a pair of needle-nose pliers.

Keeping this in view, what's the purpose of a ground wire?

The purpose of the ground wire (in U.S. terminology) is to provide a path back to the source of the electrical current — the main electrical panel — in the event of a fault. Hopefully this current is enough to trip the breaker or blow the fuse.

What happens if I don't connect the ground wire?

The appliance will operate normally without the ground wire because it is not a part of the conducting path which supplies electricity to the appliance. In the absence of the ground wire, shock hazard conditions will often not cause the breaker to trip unless the circuit has a ground fault interrupter in it.