If I am given an interval, say .51-1.49, of 100 random variables, distributed N(1,9). How would I calculate the probability that the mean of the 100 variables is in said interval. I know how I would calculate the what confidence interval this is (I'm getting that its a 12.7% confidence interval) but this doesn't mean the probability is 12.7%, right?
2026-02-26 15:12:53.1772118773
Confidence intervals and probability
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No, what you have calculated is the confidence level that the mean of your sample is within those values (i.e., you are 12.7% confident the mean is somewhere in there)
There is no way to calculate a single probability but there are two ways to calculate something similar to what you want:
To find the probability of the mean being in that interval, you need to know more about your distribution. You have $X\sim N(1,9^2)$ and a 12.7% confidence interval, $0.51\le \mu \le 1.49$, however to find the probability of you mean being in the interval you should have more information about the initial conditions of you distribution (i.e., what is your distribution describing and what are my values saying).
The first method is to calculate the interval in terms of a proportion, this will tell you what the probability of having the mean in you interval, since the proportion and the probability are the same thing in most cases.
Alternatively, when you know what you are trying to measure with this interval, you can calculate a probability of each end of the interval fulfilling you conditions.