Use the given data to construct a confidence interval for the population proportion p of the requested level. x=52, n=72, confidence level 99.9% Round the answers to at least three decimal places. The confidence interval is 00. X S
Use the given data to construct a confidence interval for the population proportion p of the requested level. x=52, n=72, confidence level 99.9% Round the answers to at least three decimal places. The confidence interval is 00. X S
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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Question
![### Constructing a Confidence Interval for a Population Proportion
To construct a confidence interval for the population proportion \( p \) at the requested level, use the given data where:
- \( x = 52 \)
- \( n = 72 \)
- Confidence level = 99.9%
#### Step: Calculate the Confidence Interval
1. **Determine sample proportion (\(\hat{p}\))**:
\[
\hat{p} = \frac{x}{n} = \frac{52}{72}
\]
2. **Find the standard error (SE)**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
3. **Look up the z-score for the 99.9% confidence level** (typically ~3.291)
4. **Calculate the margin of error (E)**:
\[
E = z \cdot SE
\]
5. **Determine the confidence interval (\(\hat{p} \pm E\))**:
\[
\text{Confidence Interval} = (\hat{p} - E, \hat{p} + E)
\]
#### Example Calculation:
- Sample proportion:
\[
\hat{p} = \frac{52}{72} = 0.722
\]
- Standard error:
\[
SE = \sqrt{\frac{0.722 \times (1 - 0.722)}{72}} \approx 0.052
\]
- Margin of error:
\[
E = 3.291 \times 0.052 \approx 0.171
\]
- Confidence Interval:
\[
(0.722 - 0.171, 0.722 + 0.171) = (0.551, 0.893)
\]
Round the answers to at least three decimal places as required:
\[
(0.551, 0.893)
\]
This interval estimates, with 99.9% confidence, that the true population proportion lies between 0.551 and 0.893.
### Diagram Explanation
**On-Screen Content Explanation**:
- **Text Content**: Provides instructions for constructing a confidence interval for the population proportion.
- **Input Box**: Allows users to enter the calculated confidence interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29a07bd8-eb1c-498d-b0a8-9eb9d6fddf09%2F9e326207-15d6-4177-aca7-90da7c6d8332%2Fo07js77_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a Confidence Interval for a Population Proportion
To construct a confidence interval for the population proportion \( p \) at the requested level, use the given data where:
- \( x = 52 \)
- \( n = 72 \)
- Confidence level = 99.9%
#### Step: Calculate the Confidence Interval
1. **Determine sample proportion (\(\hat{p}\))**:
\[
\hat{p} = \frac{x}{n} = \frac{52}{72}
\]
2. **Find the standard error (SE)**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]
3. **Look up the z-score for the 99.9% confidence level** (typically ~3.291)
4. **Calculate the margin of error (E)**:
\[
E = z \cdot SE
\]
5. **Determine the confidence interval (\(\hat{p} \pm E\))**:
\[
\text{Confidence Interval} = (\hat{p} - E, \hat{p} + E)
\]
#### Example Calculation:
- Sample proportion:
\[
\hat{p} = \frac{52}{72} = 0.722
\]
- Standard error:
\[
SE = \sqrt{\frac{0.722 \times (1 - 0.722)}{72}} \approx 0.052
\]
- Margin of error:
\[
E = 3.291 \times 0.052 \approx 0.171
\]
- Confidence Interval:
\[
(0.722 - 0.171, 0.722 + 0.171) = (0.551, 0.893)
\]
Round the answers to at least three decimal places as required:
\[
(0.551, 0.893)
\]
This interval estimates, with 99.9% confidence, that the true population proportion lies between 0.551 and 0.893.
### Diagram Explanation
**On-Screen Content Explanation**:
- **Text Content**: Provides instructions for constructing a confidence interval for the population proportion.
- **Input Box**: Allows users to enter the calculated confidence interval.
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