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.