Question

What type of barometric readings would you expect if you brought your barometer to the top of a mountain?

Answer 1

A barometer measures the atmospheric pressure.
The value of the atmospheric pressure  depends on many things, but mainly on the altitude. In fact, the value of the atmospheric pressure at a given location depends on the weight of the column of air above the location. As we go higher in altitude, the weight of this column of air decreases, and so the atmospheric pressure decreases. 
Therefore, at the top of a mountain the barometer will read a value for the atmospheric pressure lower than the value measured at sea level.