Questions: A student was asked to find a 95% 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 12.8<μ<33.8? Check all that are correct. The mean width of all widgets is between 12.8 and 33.8, 95% of the time. We know this is true because the mean of our sample is between 12.8 and 33.8. With 95% confidence, the mean width of a randomly selected widget will be between 12.8 and 33.8. There is a 95% chance that the mean of a sample of 29 widgets will be between 12.8 and 33.8. There is a 95% chance that the mean of the population is between 12.8 and 33.8. With 95% confidence, the mean width of all widgets is between 12.8 and 33.8.

A student was asked to find a 95% 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 12.8<μ<33.8?
Check all that are correct.
The mean width of all widgets is between 12.8 and 33.8, 95% of the time. We know this is true because the mean of our sample is between 12.8 and 33.8.
With 95% confidence, the mean width of a randomly selected widget will be between 12.8 and 33.8.
There is a 95% chance that the mean of a sample of 29 widgets will be between 12.8 and 33.8.
There is a 95% chance that the mean of the population is between 12.8 and 33.8.
With 95% confidence, the mean width of all widgets is between 12.8 and 33.8.

Transcript text: A student was asked to find a $95 \%$ 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 $12.8<\mu<33.8$ ? Check all that are correct. The mean width of all widgets is between 12.8 and $33.8,95 \%$ of the time. We know this is true because the mean of our sample is between 12.8 and 33.8 . With $95 \%$ confidence, the mean width of a randomly selected widget will be between 12.8 and 33.8 . There is a $95 \%$ chance that the mean of a sample of 29 widgets will be between 12.8 and 33.8 . There is a $95 \%$ chance that the mean of the population is between 12.8 and 33.8 . With $95 \%$ confidence, the mean width of all widgets is between 12.8 and 33.8 .

Solution

Solution Steps

To interpret a 95% confidence interval correctly, we need to understand that it provides a range within which we are 95% confident that the true population mean lies. It does not mean that 95% of the sample means will fall within this range, nor does it apply to individual observations. The correct interpretations should reflect this understanding.

Solution Approach

The first statement is incorrect because it misinterprets the confidence interval as applying to the sample mean rather than the population mean.
The second statement is incorrect because the confidence interval applies to the population mean, not to individual observations.
The third statement is incorrect because the confidence interval applies to the population mean, not to the sample mean.
The fourth statement is correct because it correctly interprets the confidence interval as applying to the population mean.
The fifth statement is correct because it correctly interprets the confidence interval as applying to the population mean.

Step 1: Understanding the Confidence Interval

A $ 95\% $ confidence interval for the mean width of widgets is given as $ 12.8 < \mu < 33.8 $. This means we are $ 95\% $ confident that the true population mean $ \mu $ lies within this interval.

Step 2: Evaluating the Statements

We need to evaluate the provided statements regarding the interpretation of the confidence interval:

Statement 1: Incorrect. It suggests that the mean width of all widgets is between $ 12.8 $ and $ 33.8 $ $ 95\% $ of the time, which misinterprets the confidence interval.
Statement 2: Incorrect. It implies that the interval applies to a randomly selected widget, which is not the case; the interval pertains to the population mean.
Statement 3: Incorrect. It states that there is a $ 95\% $ chance that the mean of a sample of $ 29 $ widgets will be between $ 12.8 $ and $ 33.8 $, which is a misunderstanding of the concept.
Statement 4: Correct. It accurately states that there is a $ 95\% $ chance that the mean of the population is between $ 12.8 $ and $ 33.8 $.
Statement 5: Correct. It correctly interprets that with $ 95\% $ confidence, the mean width of all widgets is between $ 12.8 $ and $ 33.8 $.

Step 3: Identifying Correct Interpretations

The correct interpretations from the statements are:

There is a $ 95\% $ chance that the mean of the population is between $ 12.8 $ and $ 33.8 $.
With $ 95\% $ confidence, the mean width of all widgets is between $ 12.8 $ and $ 33.8 $.

Final Answer

The correct interpretations are:

Statement 4: There is a $ 95\% $ chance that the mean of the population is between $ 12.8 $ and $ 33.8 $.
Statement 5: With $ 95\% $ confidence, the mean width of all widgets is between $ 12.8 $ and $ 33.8 $.

Thus, the answer is: $\boxed{\text{Statements 4 and 5 are correct.}}$

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