I am trying to estimate a function $f(x)$ at $x=0.1, 0.2, 0.3, 0.4, 0.5$. I make this estimate through some complex procedure based on a set of independent but noisy data (with known uncertainties). The outcome is a set of correlated estimates.
The null hypothesis of my study is that $f(x)=0$. I want to assess, based on any/all of these measurements, whether $f(x)\neq 0$ at some/any $x$.
I have run $N$ Monte Carlo simulations of the measurement procedure by repeating the measurement with random realizations of the data (since the measurement uncertainty is unknown). Now I have $N$ sets of estimates at these five points. I can use the $N$ trials to estimate the mean and variance of each $f(x)$ as well as the covariance between the different measurements.
Taking the mean and standard deviation across all of the $N$ trials, I find that each of the measurements are roughly $f(x)=-0.2 \pm 0.1$. At normal thresholds of significance, this would be a $2\sigma$ result and therefore not significantly different from 0. However, since all of the measurements are below zero, maybe the result is significant. However, I do not know how to assess this, given that all of the measurements are correlated.