"...Confidence interval for population mean μ:
Every time the measurements are repeated, there will be a different value for the mean of the sample. In 95% of the cases μ will be between the endpoints of the (random) confidence interval calculated from this mean, but in 5% of the cases it will not be. The actual confidence interval is calculated by entering the measured weights in the formula. Our 0.95 confidence interval becomes:
This interval has fixed endpoints, where μ might be in between (or not). There is no probability of such an event. We CANNOT say: "with probability (1 - α) the parameter μ lies in the confidence interval." We only know that by repetition in 100(1 - α) % of the cases μ will be in the calculated interval. In 100alpha % of the cases however it doesn't. And unfortunately we don't know in which of the cases this happens. That's why we say: with confidence level 100(1 - α) % μ lies in the confidence interval..." (Wikipedia)
Now I don't understand the last paragraph.
I think I am also confused about the concept of confidence interval in general. I know how to do the computations, but I just don't quite understand the meaning of it. For example, if we computed a confidence interval of (249.22,251.18) with NUMBERS (rather than random variables) as endpoints, what does it really mean?
Also, why can't we say that
(i)with probability (1 - α) the parameter μ lies in the confidence interval?
Why must we say that
(ii)by repetition in 100(1 - α) % of the cases μ will be in the calculated interval?
I can't see any difference between (i) and (ii), aren't the two statements describing the EXACT same thing? They look absolutely the same to me...
Thanks for clearing my doubts!