Interpreting what a (say) 95% confidence interval actually means is obviously tricky, especially when you are trying to teach it to students just beginning to learn stats.
One of the biggest challenges for me is that most definitions of confidence intervals actually use the concept of "confidence interval" as part of the interpretation itself. For example:
"Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value."
I understand that this definition isn't viciously circular, but it's a nightmare to try to explain to students, who naturally wonder how we can define a confidence interval as telling us what will happen if we calculate a bunch of different confidence intervals.
Frequentism is what it is, and I know that we can't technically say that (e.g.) "there is a 95% probability that the true mean lies within the bounds of the 95% CIs," but I'm wondering if there is any way to accurately define what a frequentist confidence interval means that doesn't itself refer to confidence intervals.
Based on my understanding of frequentism, I think I have an idea for such an interpretation, but I'm not at all sure it is correct.
Let's say that we are trying to estimate the population mean $\mu$ of some variable Y. We draw a random sample of N observations, and from that sample we estimate a mean $\hat \mu$ and a standard deviation $\hat \sigma$. Using the $\hat \sigma$ and N we calculate a standard error and then use that to calculate a 95% confidence bounds of A and B.
My proposed interpretation of these values is: if it were true that the true mean of Y were the $\hat \mu$ we actually estimated, and we replicated our study a 100 times, estimating the mean of Y each time, then 95% of those estimates of $\mu$ would fall between A and B.
This is clearly different from how CI's are usually defined, but based on my understanding of frequentism, sampling error, and the central limit theorem, I feel like this is a valid (and potentially more intuitive) interpretation. It is based on a conditional, but since that's something we do when interpreting p values ("if the null hypothesis were true...") it's a concept that students encounter elsewhere in statistics, and I feel it might be less confusing than the apparently circular definition used in most textbooks....assuming it's statistically accurate.
So two questions:
- Is this a statistically valid interpretation of what a confidence interval means?
- Does anyone know any other interpretation of confidence intervals that don't themselves refer to confidence intervals?
Edit: It seems the answer to #1 is "no" (although it would be great if someone could explain why that interpretation is incorrect). I also realize I should clarify that for #2, what I'm really interested is in an intuitive interpretation of what a particular, estimated CI range means (i.e. to fill in the blank in the following sentence: "I have calculated a 95% CI around an estimate that ranges from A to B, this means that __________ between A and B") that doesn't itself refer to the concepts of "confidence" or "the process of calculating confidence intervals."