Non-self-referential interpretation of confidence intervals? 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."


Does anyone know any other interpretation of confidence intervals that don't themselves refer to confidence intervals?


...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")...

Alas, not for the specific endpoints A and B that you've actually calculated for a specific CI. However, for intro students, we usually stick to CIs of the form $\hat\theta \pm \hat{MOE}$. So in the case of estimating a proportion $p$ from iid data, you could get something close to your wish by focusing on the margin of error (MOE) as the primary thing, and the CI as just a corollary of observing $\hat{p}$ and calculating the MOE.
We know that calculating MOE based on assuming $p=0.5$ gives a fixed conservative MOE for any $p$ (at the given sample size).
This MOE is fixed and known (although conservative), $p$ is fixed but unknown, and $\hat p$ is treated as random but observed. So you really can say:

*

*In 95% of iid samples of size $n$, (random) $\hat{p}$ will fall no more than (fixed) MOE away from $p$.

*In this sense, we are 95% confident that $\hat{p}$ is no more than MOE away from the true $p$.

*So this is the sense in which we are 95% confident that the interval $\hat{p}\pm MOE$ covers $p$.

Is that closer to what you want? It's not a clean direct statement about the observed endpoints (your $A$ and $B$), but it is clean and direct about the MOE.
For a somewhat-plausible scenario, tell the students to imagine they have iid data from a new political poll, in a election that looked neck-and-neck in the last poll. You're trying to estimate the proportion of voters who support candidate A and you have good reason to believe $p_A \approx 0.5$ before looking at the newest polling data, so the MOE assuming $p=0.5$ is conservative but not unreasonably so.
After building up intuition in this case where MOE can legitimately be known, you can move on to other situations where MOE isn't known and we only get an estimated $\hat{MOE}$.

*

*We cannot say that 95% of (random) $\hat\theta$s will be no more than a (fixed) $\hat{MOE}$ distance from $\theta$.

*But we can say that in 95% of samples, the random $(\hat\theta, \hat{MOE})$ pairs will be such that $\hat\theta$ is no more than $\hat{MOE}$ away from $\theta$.

*So this is the sense in which we are 95% confident that the interval $\hat{\theta}\pm \hat{MOE}$ covers $\theta$.

