A student was asked to find a $$ 99 \% $$ confidence interval for widget width using data from a random sample of size $$ n=23 $$. Which of the following is a correct interpretation of the interval $$ 11.4

Question

A student was asked to find a $$ 99 \% $$ confidence interval for widget width using data from a random sample of size $$ n=23 $$. Which of the following is a correct interpretation of the interval $$ 11.4<\mu<22.3 $$ ? Check all that are correct. A There is a $$ 99 \% $$ chance that the mean of a sample of 23 widgets will be between 11.4 and 22.3 . B With $$ 99 \% $$ confidence, the mean width of all widgets is between 11.4 and 22.3 . C There is a $$ 99 \% $$ chance that the mean of the population is between 11.4 and 22.3 . D The mean width of all widgets is between 11.4 and $$ 22.3,99 \% $$ of the time. We know this is true because the mean of our sample is between 11.4 and 22.3 . E With $$ 99 \% $$ confidence, the mean width of a randomly selected widget will be between 11.4 and 22.3 .

UpStudy AI · Accepted Answer

1. The confidence interval procedure is used to estimate the population mean $$ \mu $$.  
2. A $$99\%$$ confidence interval means that if we repeated the procedure many times, approximately $$99\%$$ of the constructed intervals would contain the true mean $$ \mu $$.  
3. Interpretation B states:  
   - "With $$ 99 \% $$ confidence, the mean width of all widgets is between $$ 11.4 $$ and $$ 22.3 $$."  
   This correctly reflects that the interval is an estimate for the true population mean $$ \mu $$ and is phrased in terms of confidence in the procedure, not probability statements about $$ \mu $$ itself (which is fixed).  
4. Evaluating the other options:  
   - Option A incorrectly refers to the mean of a sample of 23 widgets, but the confidence interval is about the population mean, not a particular sample mean.  
   - Option C incorrectly implies there is a probability statement about a fixed population mean.  
   - Option D confuses the interpretation by suggesting the sample mean is being repeatedly in between the same interval.  
   - Option E incorrectly applies the interval to individual widget widths rather than the mean of the widget width distribution.

Thus, the only correct interpretation is:

B
B: With 99% confidence, the mean width of all widgets is between 11.4 and 22.3.

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