Say I have a simple linear regression with observed values of y
y = a + bx
I then have 1000 simulations whereby I generated y randomly. (It's not a matter of reshuffling y; I generated y through a separate model.) For each simulation I have independent estimates of b. My goal is to test whether my actual estimate for b differs from my simulated b's. Normally, I would test whether b falls within the 95% confidence interval of the 1000 simulated b's. However, each b estimate has it's own standard error. How do I take this into account?
I saw one publication that gave the number of simulations that have a b +/i 95% CI that overlaps with the 95% CI of the b from the observed data. Is that the standard way of presenting the results? Also, what if I wanted to show the results (x-y relationship) graphically? What would I use for the 95% confidence interval of the slope of the simulations?