You measure 50 textbooks' weights, and find they have a mean weight of 43 ounces. Assume the population standard deviation is 11.3 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 50 textbooks' weights, and find they have a mean weight of 43 ounces. Assume the population standard deviation is 11.3 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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![**Confidence Interval Calculation for Textbook Weights**
Given Data:
- Number of textbooks (n): 50
- Sample mean (x̄): 43 ounces
- Population standard deviation (σ): 11.3 ounces
Task: Construct a 99% confidence interval for the true population mean (μ) textbook weight.
**Steps to Calculate the Confidence Interval:**
1. **Identify the Z-score for 99% Confidence Level**:
For a 99% confidence level, the Z-score (critical value) is approximately 2.576. This can be found using Z-tables or standard normal distribution calculators.
2. **Formula to Calculate the Confidence Interval**:
The confidence interval is given by:
\[
CI = \left( x̄ - Z \left( \frac{σ}{\sqrt{n}} \right), \, x̄ + Z \left( \frac{σ}{\sqrt{n}} \right) \right)
\]
where,
- \( x̄ \) is the sample mean,
- Z is the Z-score,
- \( σ \) is the population standard deviation,
- \( n \) is the sample size.
3. **Calculate the Margin of Error**:
\[
MOE = Z \left( \frac{σ}{\sqrt{n}} \right) = 2.576 \left( \frac{11.3}{\sqrt{50}} \right)
\]
4. **Compute the Interval**:
\[
CI = \left( 43 - MOE, \, 43 + MOE \right)
\]
**Final Step for Calculation**:
Calculate the Margin of Error (MOE):
\[
MOE = 2.576 \times \left( \frac{11.3}{\sqrt{50}} \right) = 2.576 \times 1.598 = 4.117
\]
So, the confidence interval is:
\[
\left( 43 - 4.12, \, 43 + 4.12 \right) = (38.88, 47.12)
\]
**Conclusion**:
Therefore, the 99% confidence interval for the true population mean textbook weight is:
\[ 38.88 < μ < 47.12 \]
**Response Fields**:
Ensure](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2539764-cbb6-4f84-96a1-bc799fc276f3%2F7cb44a52-cd30-4a85-95f3-4e844f279269%2Fbpkgpj_processed.png&w=3840&q=75)
Transcribed Image Text:**Confidence Interval Calculation for Textbook Weights**
Given Data:
- Number of textbooks (n): 50
- Sample mean (x̄): 43 ounces
- Population standard deviation (σ): 11.3 ounces
Task: Construct a 99% confidence interval for the true population mean (μ) textbook weight.
**Steps to Calculate the Confidence Interval:**
1. **Identify the Z-score for 99% Confidence Level**:
For a 99% confidence level, the Z-score (critical value) is approximately 2.576. This can be found using Z-tables or standard normal distribution calculators.
2. **Formula to Calculate the Confidence Interval**:
The confidence interval is given by:
\[
CI = \left( x̄ - Z \left( \frac{σ}{\sqrt{n}} \right), \, x̄ + Z \left( \frac{σ}{\sqrt{n}} \right) \right)
\]
where,
- \( x̄ \) is the sample mean,
- Z is the Z-score,
- \( σ \) is the population standard deviation,
- \( n \) is the sample size.
3. **Calculate the Margin of Error**:
\[
MOE = Z \left( \frac{σ}{\sqrt{n}} \right) = 2.576 \left( \frac{11.3}{\sqrt{50}} \right)
\]
4. **Compute the Interval**:
\[
CI = \left( 43 - MOE, \, 43 + MOE \right)
\]
**Final Step for Calculation**:
Calculate the Margin of Error (MOE):
\[
MOE = 2.576 \times \left( \frac{11.3}{\sqrt{50}} \right) = 2.576 \times 1.598 = 4.117
\]
So, the confidence interval is:
\[
\left( 43 - 4.12, \, 43 + 4.12 \right) = (38.88, 47.12)
\]
**Conclusion**:
Therefore, the 99% confidence interval for the true population mean textbook weight is:
\[ 38.88 < μ < 47.12 \]
**Response Fields**:
Ensure
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