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 is different to the population proportion.  The first one is a known quantity that you can compute from your sample, whereas the second one is the unknown quantity that you are making an inference about.  The second statement here is incorrect, since the sample proportion always falls within the confidence interval (by construction).
