# Confusion around Probabilistic Interpretation of Confidence Intervals

My professor told me that a confidence interval provides a measure for the resampling uncertainity. That is, if I were to re-sample this dataset a large number of times, 95% of those would contain the true parameter in this range.

To make this more probabilistically concrete, is it valid to say that a confidence interval is: the probability that the next time I re-sample, that the true parameter would fall in this interval?

• – user83346 Mar 28 '16 at 6:59
• There are several ambiguities here. If you are resampling the dataset, then the confidence intervals as computed in the resamples are giving information about the original sample but not (directly) about the population. What, then, does "true parameter" refer to: the value of the original sample or the population? Second, what does "this interval" mean at the end? The CI computed from a resample of the data or the original sample? – whuber Mar 28 '16 at 14:55
• @whuber I was taking 'resample' in the question to mean 'draw another sample from the population' rather than resampling in the sense of drawing with replacement from the sample itself. I still think that was the intent; I have edited my answer below to clarify that I take that interpretation. – Glen_b Mar 28 '16 at 15:12

## 1 Answer

Treating the parameter as fixed, you should probably rephrase slightly ("the next time I generate this interval by drawing a new sample from the population, the probability it contains the parameter...") so that it's clear that it's the interval that's the thing that's random (and also taking all the assumptions are given), but yes, with that part understood, that's one correct way to think about it.

[As whuber points out, "resample" is widely used to describe drawing a new sample from the original sample - it's often used to describe the generation of pseudo-samples in bootstrapping (which can be used to obtain approximate confidence intervals). In that case it may be possible to have it that no such resample will lead to an interval that includes the parameter. I didn't take this to be the thing being asked about.]

• Thanks Glen. That makes sense to me. Confidence intervals only quantify resampling uncertainty and can be confusing. – user46925 Mar 28 '16 at 2:30
• In light of the ambiguities in the question itself, I am concerned that this answer could be misread. It's quite possible that no CI computed from any resample of a dataset would contain the true population parameter--but this answer suggests otherwise. – whuber Mar 28 '16 at 14:57
• +1 I appreciate your clarified language, because it avoids a likely misreading of the question and the answer. To many people "resample" (especially in the phrase "resample this dataset") is a process of taking samples from the dataset rather than obtaining another dataset altogether. – whuber Mar 28 '16 at 18:56