The question asks for the correct interpretation of a 95% confidence interval $15 < \mu < 32.9$ using data from a sample size of $n = 29$.

Let's go over the options and what a confidence interval means. A 95% confidence interval means that if we were to take many samples and build a confidence interval from each one, approximately 95% of those intervals would contain the true population mean ($\mu$).

Now, let's break down each answer choice:

1. "With 95% confidence, the mean width of a randomly selected widget will be between 15 and 32.9."
   This is incorrect because a confidence interval applies to the population mean, not individual data points. The interval is about the mean of the population, not a specific widget.

2. "The mean width of all widgets is between 15 and 32.9, 95% of the time. We know this is true because the mean of our sample is between 15 and 32.9."
   This is also incorrect. A confidence interval does not guarantee that the mean will be within the interval 95% of the time. It means that we are 95% confident that the interval contains the true population mean.

3. "With 95% confidence, the mean width of all widgets is between 15 and 32.9."
   This is the correct interpretation. The confidence interval is an estimate for the population mean, and we are 95% confident that the true mean falls within this range.

4. "There is a 95% chance that the mean of the population is between 15 and 32.9."
   This is incorrect because we do not assign probability to the population mean. The population mean is fixed. Instead, the confidence interval gives us a degree of confidence about where that mean lies.

5. "There is a 95% chance that the mean of a sample of 29 widgets will be between 15 and 32.9."
   This is incorrect. The confidence interval refers to the population mean, not the mean of future samples.

Thus, the correct answer is: "With 95% confidence, the mean width of all widgets is between 15 and 32.9."

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. How do you calculate a confidence interval from sample data?
2. What happens to the confidence interval if the sample size increases?
3. How does confidence level (e.g., 95% vs. 99%) affect the width of a confidence interval?
4. Can a confidence interval ever contain the true population mean with 100% certainty?
5. How do margin of error and sample standard deviation influence confidence intervals?

Tip: Confidence intervals rely on sample data to estimate population parameters, but they are not a guarantee that the population mean is always within the range for any specific sample.