and X = 96, construct a 90% confidence interval for the population proportion, p. Give your answers to three decimals < p <
and X = 96, construct a 90% confidence interval for the population proportion, p. Give your answers to three decimals < p <
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Constructing a 90% Confidence Interval for Population Proportion**
Given:
- Sample size \( n = 120 \)
- Number of successes \( X = 96 \)
**Problem Statement:**
Construct a 90% confidence interval for the population proportion \( p \). Report the interval boundaries to three decimal places.
**Solution Steps:**
1. **Calculate the sample proportion (\( \hat{p} \))**:
\[
\hat{p} = \frac{X}{n} = \frac{96}{120} = 0.8
\]
2. **Find the critical value ( \( Z \) ) for a 90% confidence interval**:
- For a 90% confidence level, \( Z \approx 1.645 \) (This is the critical value for the standard normal distribution that corresponds to the middle 90% of the distribution.)
3. **Calculate the standard error ( \( SE \) )**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.8 \times 0.2}{120}} = \sqrt{\frac{0.16}{120}} \approx 0.0365
\]
4. **Compute the margin of error ( \( ME \) )**:
\[
ME = Z \times SE = 1.645 \times 0.0365 \approx 0.0601
\]
5. **Determine the confidence interval**:
\[
\hat{p} \pm ME = 0.8 \pm 0.0601
\]
Therefore, the confidence interval is approximately:
\[
0.8 - 0.0601 < p < 0.8 + 0.0601
\]
\[
0.740 < p < 0.860
\]
**Final Answer:**
\[
0.740 < p < 0.860
\]
**Explanation:**
- The interval [0.740, 0.860] means that we are 90% confident that the true population proportion \( p \) lies within this range based on the sample data provided.
- This interval gives a range of plausible values for the population proportion, reflecting the uncertainty inherent in using](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1a73d04-4829-40e4-8e78-0a30aa84e71e%2Fbb372571-8734-4118-a72a-d8cbf0be2bdb%2Fcamh66m_processed.png&w=3840&q=75)
Transcribed Image Text:**Constructing a 90% Confidence Interval for Population Proportion**
Given:
- Sample size \( n = 120 \)
- Number of successes \( X = 96 \)
**Problem Statement:**
Construct a 90% confidence interval for the population proportion \( p \). Report the interval boundaries to three decimal places.
**Solution Steps:**
1. **Calculate the sample proportion (\( \hat{p} \))**:
\[
\hat{p} = \frac{X}{n} = \frac{96}{120} = 0.8
\]
2. **Find the critical value ( \( Z \) ) for a 90% confidence interval**:
- For a 90% confidence level, \( Z \approx 1.645 \) (This is the critical value for the standard normal distribution that corresponds to the middle 90% of the distribution.)
3. **Calculate the standard error ( \( SE \) )**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.8 \times 0.2}{120}} = \sqrt{\frac{0.16}{120}} \approx 0.0365
\]
4. **Compute the margin of error ( \( ME \) )**:
\[
ME = Z \times SE = 1.645 \times 0.0365 \approx 0.0601
\]
5. **Determine the confidence interval**:
\[
\hat{p} \pm ME = 0.8 \pm 0.0601
\]
Therefore, the confidence interval is approximately:
\[
0.8 - 0.0601 < p < 0.8 + 0.0601
\]
\[
0.740 < p < 0.860
\]
**Final Answer:**
\[
0.740 < p < 0.860
\]
**Explanation:**
- The interval [0.740, 0.860] means that we are 90% confident that the true population proportion \( p \) lies within this range based on the sample data provided.
- This interval gives a range of plausible values for the population proportion, reflecting the uncertainty inherent in using
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