Here's the situation: I'm estimating the mean of a random variable through sampling. We know the stdev of the RV pretty well, after all that converges pretty quickly. I have a published result that says that this RV has mean 55 with a 95% confidence interval of - 20, the appropriate amount for the number of trials they did.

I'm doing my own tests, and I want to know if I am simulating this random variable correctly. I'm still running the simulation, but at the current time, I predict a mean of 44.75 with about the same confidence interval. This means the standard deviation of each of our predictions is about 10.

How can I quantitatively measure the likelihood that I am measuring the same random variable? I can plot the two normal distributions, one from each of our estimates, and say "hey, looks kinda close". I can say "eh it's only off by ten that's not too bad", but what's a good rigorous method to compare the two? Or perhaps a rule of thumb?