I think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained.
My understanding of a 90% confidence interval is: If you take 100 samples from a population to estimate a parameter, and calculate confidence intervals for each sample, then 90% of the intervals will contain the population parameter.
Let's say the parameter is difference in means between 2 populations. And for example, you took one sample (using the sample standard deviation as the population deviation is unknown) where the confidence interval is 10 +/- 1
My confusion is, what can you conclude from the confidence interval of your particular sample, since you can only conclude that 90% of intervals will contain the population mean, and each one will be different?
In other words, from your sample's interval, can you conclude whether the parameter is in it? If not, what is the point of calculating the interval?
Or is determining the interval useful because you can be reasonably sure that in other random samples, the confidence interval will be similar?
Also, what if the two populations' true difference in mean is actually 15? Then your confidence interval is correct 0% of the time, not 90%. I guess in other words, what if your sample isn't accurate at all, yet if you calculate a 90% interval, then you would still conclude 90% of the intervals would contain this parameter.