Consider the situation where there are 2 positive distributions with unknown expected values $\mu_1$ and $\mu_2$,
We have samples $S_1 = \{x_1, x_2, ..., x_n\}$ and $S_2 = \{y_1, y_2, ..., y_n\}$.
We would like to use a statistical test under the null hypothesis: $\mu_1 = k \times \mu_2$.
Do you know what kind of test should we use?
If your null is that the distributions are the same i.e. that $F_1(x) = F_2(x k)$ when you could perhaps do a permutation test: multiply the observations in sample 2 by $k$ ($x_{2i}^*=kx_{2i},\,i=1,2,...,n$ and then test whether they have the same mean, because under the null, the set of $kx_{2i}$ and $x_{1j}$ would be exchangeable.
If you are additionally interested specifically in a pure scale shift alternative (and want an easily-obtained interval for it) you might divide the first sample through by $k$ (or multiply the second by $k$), take logs and look at a location shift (whether a permutation test or a rank based test like the Wilcoxon-Mann-Whitney), before backtransforming the interval
If you have a suitable parametric assumption, using that information will generally be advantageous (power-wise), if the assumption is true or very nearly true.
Note that failure to reject doesn't mean the multiplier is $k$. You can't verify that; you might consider whether an equivalence test comes nearer to what you need in that case.
