How to test extra dimensions?

One proposed test is measuring the strength of gravity.

Gravity has the property that if there are $n$ dimensions of space, the strength of gravity when distance increases diminishes by $$frac{1}{r^{n-1}}$$ In our current understanding of the universe, there are 3 spatial dimensions, therefore the strength of gravity reduces in accordance with the well-known factor $$frac{1}{r^2}$$ But, if in case in large scale measurements we measure that factor to actually be, say for example, $$frac{1}{r^3}$$then we can deduce that there are $3 + 1 = 4$ spatial dimensions. Note that in this experiment we have to reasonably large distances for measurements.