A student was asked to find a $90 \%$ confidence interval for widget width using data from a random sample of size $n=29$. Which of the following is a correct interpretation of the interval $11.1< \mu< $ 25.6?

Check all that are correct.
The mean width of all widgets is between 11.1 and $25.6,90 \%$ of the time. We know this is true because the mean of our sample is between 11.1 and 25.6 .
With $90 \%$ confidence, the mean width of a randomly selected widget will be between 11.1 and 25.6.
There is a $90 \%$ chance that the mean of the population is between 11.1 and 25.6 .
There is a $90 \%$ chance that the mean of a sample of 29 widgets will be between 11.1 and 25.6 .
With $90 \%$ confidence, the mean width of all widgets is between 11.1 and 25.6 .
Add Work

Step 1 ：This question is about the interpretation of a confidence interval in statistics. A confidence interval is an estimated range of values which is likely to include an unknown population parameter. The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter. A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

Step 2 ：In this case, the student has calculated a 90% confidence interval for the mean width of widgets, which is between 11.1 and 25.6. This means that we are 90% confident that the true mean width of all widgets lies within this interval.

Step 3 ：The first statement is incorrect because the mean of all widgets is a fixed value and does not change. It is not correct to say that it is between 11.1 and 25.6, 90% of the time.

Step 4 ：The second statement is incorrect because the confidence interval is about the mean width of all widgets, not a randomly selected widget.

Step 5 ：The third statement is correct. This is the correct interpretation of a confidence interval.

Step 6 ：The fourth statement is incorrect because the confidence interval is about the mean width of all widgets, not a sample of 29 widgets.

Step 7 ：The fifth statement is correct. This is another way of stating the correct interpretation of a confidence interval.

Step 8 ：So, the correct interpretations are the third and fifth statements.

Step 9 ：The correct interpretations of the interval $11.1<\mu<$ 25.6 are: \n- There is a $90 \%$ chance that the mean of the population is between 11.1 and 25.6.\n- With $90 \%$ confidence, the mean width of all widgets is between 11.1 and 25.6.

Step 10 ：So, the final answer is \(\boxed{3, 5}\).