Statistical test to verify if a population has a average of k times another

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?

• I assume that you looking for a test with a specific value of $k$ in mind. Are there any assumptions about the underlying distributions, or about the variances of $S_1$ and $S_2$? – EdM Oct 19 '19 at 21:24
• O don't have an assumption about the distributions ir variances. Do you know a test in this framework that uses an assumption that is not mentioned above? – Eduardo Cesar N. Coutinho Oct 19 '19 at 21:46
• I notice you have the same sample size for both samples. Is this because there's some pairing or are the samples independent? – Glen_b Oct 20 '19 at 1:32

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
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.