The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Why?

The value of F is same for earth and the moon. Both bodies will exert the same amount of force on each other. As per universal law of gravitation, every body attracts the other body with some force and this force is same for both the bodies called gravitational force.