Question 1165661: A college statistics professor has office hours from 9:00 am to 10:30 am daily. A random sample of 20 students waiting to see the professor was selected to estimate the mean waiting time for all students. This sample produced a 95% confidence interval of 18.5 to 26.1 minutes. How should the professor interpret this interval?
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! 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.**
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## 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.
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## 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.
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