When we use the sample statistic to find a confidence interval, is there any reason to using the sample statistic after that, when we can instead refer to the confidence interval?
For example, let's say I have a sample mean of 50 and have determined a +- 5 interval with 95% confidence.
I could say "The population average is 50, plus-or-minus 5, with 95% confidence"
I could also say "The population average is between 45 and 55, with 95% confidence"
Is the probability of the population parameter occurring uniform across the confidence interval, or, would it be more likely to occur closer to the sample statistic?
The reason I ask, is, it seems to me that if the population parameter is more likely to occur closer to the sample statistic then it's useful to say "50, plus-or-minus 5", while if it's uniform then "between 45 and 55" is simpler.
