Question 1165661
The professor should interpret the $95\%$ confidence interval of $18.5$ to $26.1$ minutes as follows:

## 1. Primary Interpretation

The interval is an estimate of the true average waiting time ($\mu$) for **all students** who visit during office hours.

The correct interpretation is: **The professor can be 95% confident that the true mean waiting time ($\mu$) for all students is between $18.5$ minutes and $26.1$ minutes.**

---

## 2. Interpretation of Confidence Level

The $95\%$ confidence level refers to the **method** used to construct the interval, not the interval itself.

* If the professor were to take **many** random samples of 20 students and construct a $95\%$ confidence interval from each sample, approximately **95% of those intervals would capture the true mean waiting time** ($\mu$).
* Conversely, about $5\%$ of the intervals constructed would fail to capture the true mean.

---

## 3. What the Interval **DOES NOT** Mean

The professor should be careful to avoid common misinterpretations:

* **It does not mean** that $95\%$ of all students wait between $18.5$ and $26.1$ minutes. (The interval estimates the **mean**, not individual waiting times.)
* **It does not mean** there is a $95\%$ chance the true mean is exactly $22.3$ minutes (the midpoint). The true mean is a fixed value; it is either in the interval or it is not.