Precise and Shorthand Interpretation

Let’s return our attention to 95% confidence intervals. The precise and mathematically correct interpretation of a 95% confidence interval is a little long-winded:

• Precise interpretation: If we repeated our sampling procedure a large number of times, we expect about 95% of the resulting confidence intervals to capture the value of the population parameter.

This is what we observed. Our confidence interval construction procedure is 95% reliable. That is to say, we can expect our confidence intervals to include the true population parameter about 95% of the time.

A common but incorrect interpretation is that there’s a 95% probability that the confidence interval contains $p$. Each of the confidence intervals either does or doesn’t contain $p$. In other words, the probability is either a 1 or a 0.

So, if the 95% confidence level only relates to the reliability of the confidence interval construction procedure and not to a given confidence interval itself, what insight can be derived from a given confidence interval? For example, going back to the pennies example, we found that the percentile method 95% confidence interval for $\mu$ was (1991, 1999), whereas the standard error method 95% confidence interval was (1991, 1999). What can be said about these two intervals?

Loosely ...