I'm curious if there's a good way to estimate the probability that the true mean of a population falls closer to the observed sample mean than the tail ends of the margin of error. That is, I believe that there is unequal probability that the true mean may be greater or less than (to the right or left) of the sample mean.
But, if a survey is conducted and there is a large margin of error, how can we estimate whether it's likely that the true me is closer to the observed sample mean than at the tail end of the margin of error. That is, if the margin of error is -/+10 points, how could you calculate that there's a say 50% probability that the mean is probability -/+ 5points from the observed mean.
Ideally, I'd love a way to calculate this in R, if anyone is also familiar with that programming language.
Thank you and please let me know if i can make this question any clearer