Difference between statements about confidence intervals I am currently taking a statistics course where the following scenario comes up frequently:
Suppose a sample of size $n$ is taken from a population. $X$ is a binomial variable. The number of successes in the sample is $n\hat{p}$. The confidence interval for estimating the population proportion $p$ is:
$$
\Bigg(\hat{p} - \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}t_{.975, n-1},\; \hat{p} + \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}t_{.975, n-1}\Bigg),
$$
with $\hat{p}$ being the unbiased point estimator for the population proportion $p$.
What I don't understand is that the instructor has repeatedly emphasized the difference between these two statements, and I don't understand what that difference is:

*

*"With 95% confidence, the limits of the confidence interval contain the population proportion."

*"The sample proportion falls within the limits of the confidence interval 95% of the time"

What is the difference between these two statements?


The sample proportion falls within the limits of the confidence interval 95% of the time

It's not quite clear what this is claiming. If it's read as "95% of the times that a sample is taken and a confidence interval is calculated, the sample proportion lies within the confidence interval", that's not correct; confidence interval general are constructed to include the sample proportion (I suppose technically you could have a CI process that leads to this not being the case, but such a process would be highly nonstandard). If it means "If you continue taking more samples from the distribution that this sample came from, the sample proportion of 95% of those samples will be within the CI constructed by from this sample", that's wrong (but slightly less wrong). Confidence intervals are not calculated around the percentage of samples being in the interval, but they do tend to end up be such that they are a somewhat reasonable approximation (that is, I would expect the percentage of samples being within a percentage point of 95%).

With 95% confidence, the limits of the confidence interval contain the population proportion.

The meaning of this is also a bit fuzzy. The precise statement is that the CI was created with a process that has a 95% probability of producing a CI that includes the population proportion, and the term "confidence" is sometimes used to convey this idea, but it often results in people confusing it for "The probability of the population proportion being in the confidence interval", which is not correct.
