# Confidence Intervals - Doubts on Interpretations

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?

• 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. – Paul Sep 6 at 9:05
• 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 $\Pr\{0.5 > 0\} = 1/2$ – 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.

• 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? – Siwei Sep 7 at 6:53
• 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. – Aaron Montgomery Sep 7 at 11:25
• 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? – Siwei Sep 8 at 7:55
• 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. – Aaron Montgomery Sep 8 at 11:51