https://file.aiquickdraw.com/mathbot/file/1728649800587ukgp278e.jpg

The question asks for the correct interpretation of a 95% confidence interval \( 15 < \mu < 32.9 \) using data from a sample size of \( n = 29 \).

Let's go over the options and what a confidence interval means. A 95% confidence interval means that if we were to take many samples and build a confidence interval from each one, approximately 95% of those intervals would contain the true population mean (\( \mu \)).

Now, let's break down each answer choice:

1. **"With 95% confidence, the mean width of a randomly selected widget will be between 15 and 32.9."**  
   This is incorrect because a confidence interval applies to the population mean, not individual data points. The interval is about the mean of the population, not a specific widget.

2. **"The mean width of all widgets is between 15 and 32.9, 95% of the time. We know this is true because the mean of our sample is between 15 and 32.9."**  
   This is also incorrect. A confidence interval does not guarantee that the mean will be within the interval 95% of the time. It means that we are 95% confident that the interval contains the true population mean.

3. **"With 95% confidence, the mean width of all widgets is between 15 and 32.9."**  
   This is the correct interpretation. The confidence interval is an estimate for the population mean, and we are 95% confident that the true mean falls within this range.

4. **"There is a 95% chance that the mean of the population is between 15 and 32.9."**  
   This is incorrect because we do not assign probability to the population mean. The population mean is fixed. Instead, the confidence interval gives us a degree of confidence about where that mean lies.

5. **"There is a 95% chance that the mean of a sample of 29 widgets will be between 15 and 32.9."**  
   This is incorrect. The confidence interval refers to the population mean, not the mean of future samples.

Thus, the correct answer is: **"With 95% confidence, the mean width of all widgets is between 15 and 32.9."**

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you calculate a confidence interval from sample data?
2. What happens to the confidence interval if the sample size increases?
3. How does confidence level (e.g., 95% vs. 99%) affect the width of a confidence interval?
4. Can a confidence interval ever contain the true population mean with 100% certainty?
5. How do margin of error and sample standard deviation influence confidence intervals?

**Tip:** Confidence intervals rely on sample data to estimate population parameters, but they are not a guarantee that the population mean is always within the range for any specific sample.

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 15 < μ < 32.9?

Learn how to interpret a 95% confidence interval for widget widths. This problem involves key statistical concepts such as confidence intervals, sample and population means, and the central limit theorem. Suitable for Grades 11-12 or introductory college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation