To interpret a 90% confidence interval correctly, we need to understand that this interval provides an estimate for the population mean, not individual measurements. The interval $11 < \mu < 29.6$ means we are 90% confident that the true population mean $\mu$ of the widget width lies within the range 11 to 29.6.

Let's evaluate each option:

1. "The mean width of all widgets is between 11 and 29.6, 90% of the time. We know this is true because the mean of our sample is between 11 and 29.6."

   • Incorrect. A confidence interval does not guarantee that all widget widths are within this range, only that the population mean lies within this interval with 90% confidence.

2. "With 90% confidence, the mean width of a randomly selected widget will not be between 11 and 29.6."

   • Incorrect. Confidence intervals apply to the population mean, not individual widgets.

3. "There is a 90% chance that the mean of a sample of 25 widgets will be between 11 and 29.6."

   • Incorrect. This interpretation is for the population mean, not the mean of another sample of widgets.

4. "With 90% confidence, the mean width of all widgets is between 11 and 29.6."

   • Correct. This correctly interprets that we are 90% confident the true population mean falls within the interval.

5. "There is a 90% chance that the mean of the population is below 11."

   • Incorrect. This is irrelevant, as the interval provides information about the range $11 < \mu < 29.6$, not values below 11.

Answer: The correct option is the fourth one: "With 90% confidence, the mean width of all widgets is between 11 and 29.6."

Would you like further clarification on confidence intervals?

Follow-up Questions:

1. What is the difference between confidence intervals and probability for individual data points?
2. How would a 95% confidence interval differ from a 90% confidence interval?
3. Can confidence intervals ever guarantee the true mean of the population?
4. How is the sample size related to the width of a confidence interval?
5. What does it mean if a confidence interval is very wide?

Tip: Confidence intervals provide a range for where the population mean is likely to be, not individual values.