Many times I have heard the physicist Michio Kaku saying that the deviation on Newton's inverse law could demonstrate the existence of multidimensional space, which could support one of the aspect of string theory. I also know that the general expression shows that magnitude of $F$ is proportional to $\frac{1}{r^{n-1}}$, or:
$$ F \propto \frac{1}{r^{n-1}} $$
My question is: How do we get this deviation experimentally? How do we know, making the experiment that the deviation is caused by the interation of gravity with another dimension, not because of the measurement devices?
