Suppose we use a sample mean $\bar{X}$ to construct a $95\%$ confidence interval $[a, b]$.

I was told that it is incorrect to say that there is a $95\%$ probability that the population mean lies between $a$ and $b$. Because the population mean is a constant and not a random variable. The probability that a constant falls within any given range is either $0$ or $1$.

However, from the textbook it is said that we expect $95\%$ of the confidence intervals to include the population mean.

If $95\%$ of the confidence intervals are expected to include the population mean, then each confidence interval has $95\%$ probability to include the population mean. Therefore I think it is correct to say there is a $95\%$ probability that the population mean lies between $a$ and $b$.

Where did I make mistakes?

  • $\begingroup$ The random element here is the confidence interval. The sample mean is a random variable and each instance of it generates a random interval. 95% of these random intervals will contain the true mean (in the frequentist sense - generate a large number of samples and their intervals etc). The true mean is not going anywhere but, potentially, 5% of the confidence intervals will not capture it. To say "95% probability that the population mean lies between a and b" begs the question of why that particular a and b is so special. $\endgroup$ – Paul Sep 6 at 9:05
  • $\begingroup$ As what Paul said above, you are using a pair of estimator as an interval estimator, to cover a deterministic but unknown parameter of interest. The estimators are random variable before they realize into $a, b$. The realizations $a, b$ are no longer random, and usually we called them the estimate. E.g., for a normal random variable $Z$ with mean $0$, we know that $\Pr\{Z > 0\} = 1/2$. Suppose you observe one realization of $Z$ as $0.5$. Then you will not say $\Pr\{0.5 > 0\} = 1/2$ $\endgroup$ – BGM Sep 6 at 10:47

The two comments on this question are good. I'll try to flesh them out a bit more into an answer.

You're right that the interpretation of 95% confidence is as follows: if you collected many samples, and from each one generated a different confidence interval, then 95% of the intervals generated would capture the true mean $\mu$ inside them.

So, why is the other interpretation incorrect? It's tricky to see, in part because of the use of the placeholders $a, b$. Let's make this concrete and suppose your 95% interval for $\mu$ is specifically $[2.6, 8.3]$. If someone asked you what the probability that $\mu$ was in $[2.6, 8.3]$ was, your answer should be: "That question makes no sense." ** Remember that $\mu$ is a fixed number, and you just don't have the privilege of knowing which specific number it is. You would never ask something like the probability that $2$ is in the interval $[2.6, 8.3]$, or the probability that $\pi = 3.14159...$ is in the interval $[2.6, 8.3]$. Either the numbers are in the interval, or they aren't.

That's the issue with the interpretation at the top of the question. Once you actually commit to a sample and its resulting confidence interval, it either has $\mu$ inside it, or it doesn't. But the endpoints of the interval are no longer random variables (because you've realized them into actual numbers), and $\mu$ was never a random variable in the first place. It's easy to obfuscate this nuance when you use placeholders like $a, b$ for the endpoints of the interval.

**I'll add the obligatory note: this is all the "frequentist" perspective of statistics. If you want to think of $\mu$ as a random variable that can change, you can do that too. It's called "Bayesian" statistics, but it's usually not taught at introductory levels.

  • $\begingroup$ Suppose I obtain a $95\%$ confidence interval $[0.8, 0.9]$ and we can't say this specific interval has $95\%$ probability to contain $\mu$. So what is the meaning of this specific interval? $\endgroup$ – Siwei Sep 7 at 6:53
  • $\begingroup$ If you repeated the process of constructing the interval many times (including collecting new samples), 95% of the intervals you derived would hav the true parameter inside them. $\endgroup$ – Aaron Montgomery Sep 7 at 11:25
  • $\begingroup$ Yes, that is right. But in real world we can only collect a few samples to construct confidence intervals due to costs. Suppose we collect ten samples to construct ten $95\%$ confidence intervals $[a_i, b_i]$ where $i=1, 2, 3, ..., 10$. What information can we get from those ten intervals? $\endgroup$ – Siwei Sep 8 at 7:55
  • $\begingroup$ Same answer as above; 95% of the intervals generated in this way would contain the parameter. But once you're looking at a particular instantiated 10 of them, the probabilistic interpretation vanishes. $\endgroup$ – Aaron Montgomery Sep 8 at 11:51

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