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The question asks for the correct interpretation of a 95% confidence interval for the mean width of a widget, given as \(10.7 < \mu < 34.5\) for a random sample of size \(n = 22\). 

Here are explanations for each option:

1. **Incorrect**: "With 95% confidence, the mean width of a randomly selected widget will be between 10.7 and 34.5."
   - This statement is wrong because confidence intervals apply to the population mean, not to individual data points (widgets).

2. **Incorrect**: "There is a 95% chance that the mean of a sample of 22 widgets will be between 10.7 and 34.5."
   - This statement is not correct because the confidence interval is about the population mean, not the mean of future samples.

3. **Incorrect**: "The mean width of all widgets is between 10.7 and 34.5, 95% of the time. We know this is true because the mean of our sample is between 10.7 and 34.5."
   - This statement implies certainty about the population mean, which is incorrect. A confidence interval does not guarantee that the mean of all widgets lies in the interval.

4. **Correct**: "There is a 95% chance that the mean of the population is between 10.7 and 34.5."
   - This is the correct interpretation. The 95% confidence interval means we are 95% confident that the true population mean falls within the range 10.7 to 34.5.

5. **Incorrect**: "With 95% confidence, the mean width of all widgets is between 10.7 and 34.5."
   - This statement is slightly misleading. It implies the mean of all widgets definitely falls in this range, whereas the interval gives a 95% confidence that the population mean is within that range.

So, the correct answer is the fourth option: **"There is a 95% chance that the mean of the population is between 10.7 and 34.5."**

Do you want more details or have any further questions?

Here are 5 related questions:
1. What assumptions are required to construct a confidence interval?
2. How does sample size affect the width of a confidence interval?
3. What does increasing the confidence level (e.g., from 95% to 99%) do to the interval width?
4. Can we use confidence intervals to make inferences about individual data points?
5. How is a confidence interval constructed for proportions instead of means?

**Tip**: A larger sample size generally leads to a narrower confidence interval because it reduces the standard error of the mean.

A student was asked to find a 95% confidence interval for widget width using data from a random sample of size n = 22. Which of the following is a correct interpretation of the interval 10.7 < μ < 34.5?

Learn how to interpret a 95% confidence interval (10.7 < μ < 34.5) for widget width using a sample size of 22. This problem covers key statistical concepts, including confidence intervals, sampling, and the application of the Central Limit Theorem.

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