2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.