A student was asked to find a $99 \%$ confidence interval for widget width using data from a random sample of size $n=25$. Which of the following is a correct interpretation of the interval $12.8<\mu<31.2$ ? Check all that are correct. There is a 99\% chance that the mean of a sample of 25 widgets will be between 12.8 and 31.2 . There is a $99 \%$ chance that the mean of the population is between 12.8 and 31.2 . With $99 \%$ confidence, the mean width of all widgets is between 12.8 and 31.2 . With $99 \%$ confidence, the mean width of a randomly selected widget will be between 12.8 and 31.2 . The mean width of all widgets is between 12.8 and $31.2,99 \%$ of the time. We know this is true because the mean of our sample is between 12.8 and 31.2 .
Solution
Step 1 ：The question is asking for the correct interpretation of a confidence interval. A confidence interval is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. In this case, the population parameter is the mean width of the widgets.
Step 2 ：The correct interpretation of a confidence interval is that we are 99% confident that the true population mean lies within the interval. This does not mean that there is a 99% chance that the mean of a sample of 25 widgets will be between 12.8 and 31.2, because the sample mean can vary from sample to sample.
Step 3 ：Also, it does not mean that there is a 99% chance that the mean of the population is between 12.8 and 31.2, because the population mean is a fixed value and does not have a probability associated with it.
Step 4 ：It does not mean that the mean width of a randomly selected widget will be between 12.8 and 31.2, because the confidence interval is about the population mean, not individual measurements.
Step 5 ：Finally, it does not mean that the mean width of all widgets is between 12.8 and 31.2, 99% of the time, because the confidence interval is about the population mean, not the mean of our sample.
Step 6 ：Final Answer: \(\boxed{\text{With 99\% confidence, the mean width of all widgets is between 12.8 and 31.2}}\)
