How to account for uncertainty in population "mu" in one-sample Wilcox test As a toy example, I have measurements from a sample of classes, measuring the proportion of black students in those classes. I wish to compare this to the proportion of black children in the US to determine whether classes tend to have a greater or lesser proportion of black students than would be expected from their population.
Assuming black children make up 15% of all children in the US, I imagine setting up a one-sample wilcox test like such:
classes_measures <- c(0.1, 0.1, 0.2, 0.2, 0.3, 0.3) 

wilcox.test(x = classes_measures, mu = 0.15, alternative = 'two.sided')

The trouble with this is that the measure of black children is itself drawn from a sample, and when population estimates are published by the census, I can obtain both the estimated value and the standard error for that estimate.
Ideally, I would like to perform a wilcox test for which the theoretical value mu also has a standard error, such that if the error is quite large, the probability of our wilcox test being significant is much less. Is there a way to accomplish this? Is this fundamentally unworkable because they're not comparing the same types of values (one is a sample of class rooms whereas the other is a population estimate of students). Theoretically it doesn't seem that it should matter for the Wilcox procedure where the theoretical value comes from, it is merely the value used to compare with the sample values...

As far as I know, the Wilcoxon rank-sum test won't handle that. However, you could try using Welch's t-test. Yes, this isn't normally distributed data, but Welch's t is fairly robust, and you have the central limit theorem in your side.
The inputs $\bar x_1$ and $\bar x_2$ would be the sample and census values respectively, and the variances $s^2_{\bar x_1}$ and $s^2_{\bar x_2}$ can also be obtained from the sample and census.
The degrees of freedom will require a measure of sample size, hopefully which you can obtain for both. I suspect $n_1$ will have classes as units and $n_2$ will have children as units.
You'd have to code this by hand, I doubt any statistical package would have this included.
