Use the given data to construct a confidence interval of the requested level. X = 178, N=531, Confidence level 90%
Use the given data to construct a confidence interval of the requested level. X = 178, N=531, Confidence level 90%
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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![### Constructing a Confidence Interval
**Problem Statement:**
Use the given data to construct a confidence interval of the requested level.
**Given Data:**
- \( x = 178 \)
- \( N = 531 \)
- Confidence Level: 90%
### Instructions:
1. **Identify the Sample Proportion (\(\hat{p}\))**:
- The sample proportion \(\hat{p}\) can be calculated using the formula \(\hat{p} = \frac{x}{N}\).
- Here, \( x \) is the number of successes, and \( N \) is the sample size.
2. **Calculate the Sample Proportion**:
- \(\hat{p} = \frac{178}{531}\)
- \(\hat{p} \approx 0.3354\)
3. **Determine the Z-Score for the Confidence Level**:
- For a 90% confidence level, the Z-score is approximately 1.645 (this value can be found in Z-tables).
4. **Calculate the Standard Error (SE)**:
- The Standard Error (SE) is calculated using the formula \(SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{N}}\).
- Here, \( \hat{p} \) is the sample proportion, and \( N \) is the sample size.
\[SE = \sqrt{\frac{0.3354 \times (1 - 0.3354)}{531}} \approx 0.0205\]
5. **Calculate the Margin of Error (ME)**:
- The Margin of Error (ME) is calculated using the formula \(ME = Z \times SE\).
- Here, \( Z \) is the Z-score corresponding to the desired confidence level.
\[ME = 1.645 \times 0.0205 \approx 0.0337\]
6. **Determine the Confidence Interval**:
- The confidence interval (CI) is given by \(\hat{p} \pm ME\).
\[
CI = 0.3354 \pm 0.0337
\]
- So the interval is approximately (0.3017, 0.3691).
### Conclusion:
The 90% confidence interval for the given data is approximately (0.3017](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2561c632-65dd-4d60-9cc7-c4695e3617a7%2F667f1b15-b84d-4ee7-8bfe-e01a733027b8%2Fgvcytv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a Confidence Interval
**Problem Statement:**
Use the given data to construct a confidence interval of the requested level.
**Given Data:**
- \( x = 178 \)
- \( N = 531 \)
- Confidence Level: 90%
### Instructions:
1. **Identify the Sample Proportion (\(\hat{p}\))**:
- The sample proportion \(\hat{p}\) can be calculated using the formula \(\hat{p} = \frac{x}{N}\).
- Here, \( x \) is the number of successes, and \( N \) is the sample size.
2. **Calculate the Sample Proportion**:
- \(\hat{p} = \frac{178}{531}\)
- \(\hat{p} \approx 0.3354\)
3. **Determine the Z-Score for the Confidence Level**:
- For a 90% confidence level, the Z-score is approximately 1.645 (this value can be found in Z-tables).
4. **Calculate the Standard Error (SE)**:
- The Standard Error (SE) is calculated using the formula \(SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{N}}\).
- Here, \( \hat{p} \) is the sample proportion, and \( N \) is the sample size.
\[SE = \sqrt{\frac{0.3354 \times (1 - 0.3354)}{531}} \approx 0.0205\]
5. **Calculate the Margin of Error (ME)**:
- The Margin of Error (ME) is calculated using the formula \(ME = Z \times SE\).
- Here, \( Z \) is the Z-score corresponding to the desired confidence level.
\[ME = 1.645 \times 0.0205 \approx 0.0337\]
6. **Determine the Confidence Interval**:
- The confidence interval (CI) is given by \(\hat{p} \pm ME\).
\[
CI = 0.3354 \pm 0.0337
\]
- So the interval is approximately (0.3017, 0.3691).
### Conclusion:
The 90% confidence interval for the given data is approximately (0.3017
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