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From Wikipedia

A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained.

Many other sources (including this cv question) says the same.

But, from the selected answer of this question, it seems, for e.g., a 95% C.I. for mean indeed suggests that the true mean will be in the CI 95% of the time. To quote:

This means that the unknown mean of all baskets $\mu$ is (with a probability of $95\%$) in the interval $[\bar{w}-1.96\frac{\sigma}{\sqrt{n}};\bar{w}+1.96\frac{\sigma}{\sqrt{n}}]$

And how large the CI indicates the precise our estimation is, which makes sense.

But these two sources contradict. If I understand correctly, it might be the case that the statement in the selected answer above is valid up until we calculate the C.I., as it is still a random variable. But once estimated, both the true mean and the CI are fixed, and the probabilistic statement is not appropriate anymore.

If this is correct, then my question is, once we estimate CI, what things we can say about it? How/why does it matter whether the CI is large or small, since we cannot say if true mean falls in it or not?

Rakib
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    Many frequentist statisticians object to the use of the word probability attached to an unknown fixed parameter value. For a hypothetical sample of size $n$ from a normal dist'n with $\sigma$ known, $\mu$ unknown, they are happy enough to write $P(-1.96 < \frac{\bar W - \mu}{\sigma/\sqrt{n}} < 1.96),$ but once $\bar W$ is observed they prefer 'confidence' when referring to the interval estimate $\bar W \pm 1.96\sigma/\sqrt{n}.$ They say, "Either $\mu$ lies in the interval or it doesn't. No _probability_ about it. _Process_ gives useful info in 95% of cases over the long run." – BruceET Sep 19 '18 at 05:09
  • Maybe another way I can phrase it is this: 95% of _what_ time? In this case, 95% of experiments will yield a confidence interval containing the mean. That's not the same meaning as the first paragraph you quoted, which suggests that 95% of "the time" you see a confidence interval + data exactly equal to what you have, the true mean will be in said interval as opposed to another interval. – Kevin Sep 19 '18 at 05:37
  • @BruceET, can you please elaborate "Process gives useful info in 95% of cases over the long run"? – Rakib Sep 19 '18 at 13:50
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    To a frequentist, the CI is a statement about the data. The CI arises from a probability comp. with prob. 95% of being correct. In an long seq, of independent trials (datasets) it will be correct in 95% of them. Sign on frequentist consulting statistician's door: "I use 95% confidence intervals; I lie to 5% of my clients." // Bayesian approach is to get prior distributions from clients that describe their view of situation in which data collected. Then data and prior together give posterior, used to make inferences about situation. // Neither approach fully satisfactory. Hence arguments. – BruceET Sep 19 '18 at 15:41
  • @BruceET, thanks. But still is there anything concrete that we can say once CI is estimated, or how CI can be useful to asses the quality of our sample if nothing can be said (From either frequentist or Bayesian perspective)? – Rakib Sep 19 '18 at 16:13
  • Limited-character comments not ideal for informative discussion. Both frequentist CIs and Bayesian probability intervals can give info useful in practice. Interval $(95,103)$ for $\mu$ gives you an idea $\mu$ is near 99 and you shouldn't worry its much < 95 or > 103 (although there's still some chance of that). // Statistics deals with inductive reasoning not deductive reasoning. So (as in real life) almost nothing important is 100% sure. If you want iron-clad logical certainty, consult a mathematician who never makes errors in logic. (But not with data collected in imperfect circumstances.) – BruceET Sep 19 '18 at 17:33

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