Error/uncertainty interval goes negative while the value is known to be positive, how to report the interval? Say we measure the coefficient of friction between two materials to be $0.03 \pm 0.05$ where we have used the formula $\Delta Y =  \sqrt{ \left ( \frac{\partial F}{\partial X_1} \right )^2 (\Delta X_1)^2 +...+ \left (\frac{\partial F}{\partial X_N} \right )^2 (\Delta X_N)^2}$ where $Y = F(X_1, ..., X_N)$ in order to compute the error.
If we assume conservation of energy holds, then we know that the coefficient of friction cannot be negative. So I am tempted to think the error of the COF should be reported as $0.03 +0.05/-0.03$.
However when writing this question on PSE I realized what "the error" is (say 1 standard deviation) and thus it should be reported as the formula suggests, namely $\pm 0.05$. Because it does not imply the value could be negative any more than if we had obtained say $0.93 \pm 0.05$. Well maybe more probable indeed, but the point is that in both cases there is some probability for the COF to be negative, since the upper and lower bounds don't mean the value measured is necessarily restrained inside of them.
I'd like to know whether I'm right or off. 
Edit: Some people are suggesting me to reduce the uncertainty/error to bypass this "problem" or just to get a better evaluation, by making more and/or better measurements. This is not my goal. Say I measured the quantity above with that particular interval and say I cannot reproduce the experiment for some reason. Note that in reality I don't have to report such a value with such an uncertainty, I am just curious on how to deal with such a case.
 A: When you report the uncertainty in your measurement you basically state "this measurement could have been obtained with the underlying values of X in this range".
That is not the same as saying "X can have any of these values". If you actually want to give a confidence interval you could say something like "there is 95% confidence that X is in the range [0, y]". But in that case, especially with the numbers you give, you might have to deal with the asymmetry of the situation (the interval is no longer +- 1.96 $\sigma$.)
I am not aware of a uniform convention for this case. When in doubt use words to clarify - compared to the effort of the measurement, writing a few words to communicate unambiguously is well worth it.
A: The first-order law of propagation of uncertainty that you have reported cannot deal in a coherent way with asymmetric distributions or physical limits like those of your example.
This means that if you want to have a better uncertainty evaluation, taking into account physical limits in a realistic way, you should employ more refined methods of propagation. Just truncating the uncertainty interval is generally considered a poor approach.
In particular, you can consider the Monte Carlo method (see, e.g., this guide) or a Bayesian evaluation. For the latter, refer to a book on Bayesian statistics.
A: My take on my own question:
The uncertainty should be reported as $\pm 0.05$ in your case, because you used the right formula to compute the uncertainty based on the fact that $Y=F(X_1, ..., X_N)$. 
Note that you're not implying that the value of the COF has to lie in the uncertainty interval, because that interval is just worth 2 sigma. So, if you had a COF of say $0.93$ with an uncertainty of $0.05$, even though values near 0 are well outside the uncertainty range, there's still a non zero probability that the COF is actually near 0 even though that probability is in practice indistinguishable from 0. 
Now the uncertainty you calculated for such a value of the COF may seem "too big" but that is purely subjective and beside the point of the original question.
A: One workaround whenever one has a manifestly positive quantity $\mu>0$ is to report the variable $\ln\mu\in \mathbb{R}$ instead, which is then manifestly real, and hence doesn't have the problem with unphysical error intervals. Another is to use lopsided error intervals.
