# Are these statements about Confidence Interval correct?

These are statements I got from a pile of homework my colleagues have submitted for a statistics course

1. "...(if) mean is contained in the confidence interval, we can tell that there is a 95% probability of that the results are not significantly different from the population mean (null hypothesis)..."

2. "...CI tells you via raw data an interval where mean value is located at certain probability..."

I think these statements are wrong due to the fact that we cannot make probabilistic statements about means regarding confidence intervals.

For statement 1, it is not explicitly mentioned whether he is referring to the population mean, or the sample mean. Either way, I think he should've said "...with 95% confidence level that the result is not significantly different from the population mean..."

For statement 2, it is my understanding that neither population mean nor sample mean cannot be located at any interval at certain probability, since they're pretty much fixed values (for the sample mean, he's referring to the sample mean already acquired via data). To use the probability statement, I think he should've stated that "95% CI, once calculated, has 95% probability that it will include the population mean"

• I would say that statement 2 is wrong "on a case by case basis (for specific cases) you can not really say that the probability for p inside the interval is 95%" (stats.stackexchange.com/a/297237/164061). I would say that statement 1 (which is not general but about the context with CI for the mean specifically) that it is almost correct. Confidence intervals are a bit like hypothesis tests ( stats.stackexchange.com/questions/351320/… ) but statement 1 does not express the hypothesis testing well. – Martijn Weterings Oct 8 '18 at 5:44

• I am not saying that you are not right about comment 1 but regarding the statement "The sample mean is always in the confidence interval." one could say a few things: When you pick a sample from a uniform distribution $U(0,c)$ then a confidence interval for the population mean may not need to contain the sample mean. E.g when the sample is a single number $x$ then the $\alpha$ confidence interval for the mean is $(\frac{1}{2}c, \frac{1}{2(1-\alpha)} c)$, and for confidence intervals <50% the confidence interval for the population mean will certainly not contain the sample mean. – Martijn Weterings Oct 8 '18 at 8:45