PA-28 Flight Instructor reported engine malfunction/failure during final approach.

Narrative

Power loss on base to final. We took off with a full right tank and about a half full left tank for a total of about 34 gallons of fuel. I figured the student before us just never swapped tanks. So we started on the full tank (right); and I told my student we would stay on that one for the whole flight. As we were coming back in to land; we encountered engine roughness on base and eventually a total loss of power on base to final. I took controls and notified tower on final that we lost power. A couple seconds later; I told them that we wouldn't make the runway and would be putting it down in the field just prior to the runway. I dipped the nose to maintain Vg and we suddenly gained some power back. I notified Tower that we would make the runway; landed; and made it back to the ramp. The engine died completely on the ramp before we made it to parking. When I shut the plane off and got out to inspect; our left tank was completely depleted and the right was full. I believed and verified with our A&P that the fuel selector was not working. It showed that we were pulling from the right tank; but it was pulling from the left.