For you to weigh 1/16 of your current weight, the force of attraction between you and the earth must be decreased to 1/16 of its current value. That would mean that you would have to be 4 times as far away from the center of the earth as you are now (because gravitational force is inversely proportional to 1/r²). This makes sense when you consider the equation for gravitational force F = Gm₁m₂/r². G, m₁, and m₂ do not change, so an increase in r by a factor of four would mean a decrease in F by factor of 4² = 16.
Since the radius of earth is 6400 km, you need to be 6400 * 4 = 25600km away from the center of the earth, or 25600 - 6400 = 19200km above the surface of the earth.