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?
Best Answer
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.