I have a population (n=5000) and I know everything about it (all point values, mean, standard deviation, etc.) From this, I will sample 1000 items. I can calculate the mean of the sample and compare it to the population mean. Oh, and the first assumption (truly a fact) is that I am an armchair statistician, not at all trained.
I want to generalize an answer to this question: What is the probability a random sample mean is expected to be more than x% different from the population mean some percentage (say, 50%) of the time, given I know everything about the population?
What I've Tried
I did a brute force analysis of one population. I took 100x 1000-item samples, calculated the mean of each sample, and looked at the distribution of these means. I observed, as expected (I think), that the differences between sample and population means is roughly normally distributed around zero. So, with 100 trials analyzed I was able to say random sampling of this particular population will produce sample means more than +/-5% different from the population mean 50% of the time due to chance. (Is that a correct conclusion?)
I also looked at some related questions here but I did not find quite what I was looking for. I get that the mean of the sample means should converge to the population mean, but I can't quite make the leap to understand how the sample means are distributed around the population's mean.
Other Possibly Useful Information
- The population's variable of interest is always right-skewed (and possibly has a lognormal distribution)
- All population variable values are positive
- The population's standard deviation is typically 2-5 times the population's mean