You measure 20 randomly selected textbooks' weights, and find they have a mean weight of 37 ounces. Assume the population standard deviation is 8.1 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight. Give your answers as decimals, to two places
You measure 20 randomly selected textbooks' weights, and find they have a mean weight of 37 ounces. Assume the population standard deviation is 8.1 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight. Give your answers as decimals, to two places
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Calculating a 99% Confidence Interval for Textbook Weights
**Given Data:**
- **Sample Size (n):** 20 textbooks
- **Sample Mean (\(\bar{x}\)):** 37 ounces
- **Population Standard Deviation (\(\sigma\)):** 8.1 ounces
### Objective
We aim to construct a 99% confidence interval for the true mean weight (\(\mu\)) of textbooks.
### Steps to Calculate the Confidence Interval:
1. **Determine the Z-score for 99% Confidence Level:**
For a 99% confidence level, the Z-score is approximately 2.576 (from standard Z-tables).
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{8.1}{\sqrt{20}}
\]
3. **Calculate the Margin of Error (ME):**
\[
\text{ME} = Z \times \text{SEM} = 2.576 \times \frac{8.1}{\sqrt{20}}
\]
4. **Compute the Confidence Interval:**
\[
\text{Lower Limit} = \bar{x} - \text{ME}
\]
\[
\text{Upper Limit} = \bar{x} + \text{ME}
\]
### Final Confidence Interval:
Given the steps above, you can plug in the values to get the confidence interval.
### Answer Format:
Enter your answers as decimals rounded to two places.
\[
\boxed{\text{Lower Limit}} < \mu < \boxed{\text{Upper Limit}}
\]
Ensure your answer boxes are filled accordingly with your final results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa28b18f-3e2d-4592-9d09-7b963ee49630%2F8dae0bf4-87ea-489e-969d-5e37f575f5de%2Fyqsagv7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating a 99% Confidence Interval for Textbook Weights
**Given Data:**
- **Sample Size (n):** 20 textbooks
- **Sample Mean (\(\bar{x}\)):** 37 ounces
- **Population Standard Deviation (\(\sigma\)):** 8.1 ounces
### Objective
We aim to construct a 99% confidence interval for the true mean weight (\(\mu\)) of textbooks.
### Steps to Calculate the Confidence Interval:
1. **Determine the Z-score for 99% Confidence Level:**
For a 99% confidence level, the Z-score is approximately 2.576 (from standard Z-tables).
2. **Calculate the Standard Error of the Mean (SEM):**
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{8.1}{\sqrt{20}}
\]
3. **Calculate the Margin of Error (ME):**
\[
\text{ME} = Z \times \text{SEM} = 2.576 \times \frac{8.1}{\sqrt{20}}
\]
4. **Compute the Confidence Interval:**
\[
\text{Lower Limit} = \bar{x} - \text{ME}
\]
\[
\text{Upper Limit} = \bar{x} + \text{ME}
\]
### Final Confidence Interval:
Given the steps above, you can plug in the values to get the confidence interval.
### Answer Format:
Enter your answers as decimals rounded to two places.
\[
\boxed{\text{Lower Limit}} < \mu < \boxed{\text{Upper Limit}}
\]
Ensure your answer boxes are filled accordingly with your final results.
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