X₁ = 369, n₁ = 508, X₂ = 426, n₂ = 599, 95% confidence *** The researchers are confident the difference between the two population proportions, P₁-P₂, is between and. [ (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)
X₁ = 369, n₁ = 508, X₂ = 426, n₂ = 599, 95% confidence *** The researchers are confident the difference between the two population proportions, P₁-P₂, is between and. [ (Use ascending order. Type an integer or decimal rounded to three decimal places as needed.)
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
![**Construct a Confidence Interval for Population Proportion Differences**
**Objective:** Construct a confidence interval for \( p_1 - p_2 \) at a given level of confidence.
Given data:
- \( x_1 = 369 \)
- \( n_1 = 508 \)
- \( x_2 = 426 \)
- \( n_2 = 599 \)
Confidence level: 95%
**Instructions for Solution:**
The researchers aim to determine the confidence interval for the difference between two population proportions, \( p_1 - p_2 \).
**Steps:**
1. Calculate the sample proportions:
\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{369}{508}
\]
\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{426}{599}
\]
2. Determine the standard error (SE) for the difference of the proportions:
\[
SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}
\]
3. Find the Z-value corresponding to a 95% confidence level (usually Z=1.96).
4. Calculate the margin of error (ME):
\[
ME = Z \times SE
\]
5. Determine the confidence interval:
\[
(\hat{p}_1 - \hat{p}_2) \pm ME
\]
**Conclusion:** Fill in the confidence interval values calculated above:
The researchers are [ ]% confident the difference between the two population proportions, \( p_1 - p_2 \), is between [ ] and [ ].
**Note:**
Make sure to use ascending order for the interval and round to three decimal places as needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb999064-f838-4c3f-8d53-1a36dcc65fdf%2Fed87bfe2-252e-4267-b95a-1596959d8831%2Fnsulhvh_processed.png&w=3840&q=75)
Transcribed Image Text:**Construct a Confidence Interval for Population Proportion Differences**
**Objective:** Construct a confidence interval for \( p_1 - p_2 \) at a given level of confidence.
Given data:
- \( x_1 = 369 \)
- \( n_1 = 508 \)
- \( x_2 = 426 \)
- \( n_2 = 599 \)
Confidence level: 95%
**Instructions for Solution:**
The researchers aim to determine the confidence interval for the difference between two population proportions, \( p_1 - p_2 \).
**Steps:**
1. Calculate the sample proportions:
\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{369}{508}
\]
\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{426}{599}
\]
2. Determine the standard error (SE) for the difference of the proportions:
\[
SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}
\]
3. Find the Z-value corresponding to a 95% confidence level (usually Z=1.96).
4. Calculate the margin of error (ME):
\[
ME = Z \times SE
\]
5. Determine the confidence interval:
\[
(\hat{p}_1 - \hat{p}_2) \pm ME
\]
**Conclusion:** Fill in the confidence interval values calculated above:
The researchers are [ ]% confident the difference between the two population proportions, \( p_1 - p_2 \), is between [ ] and [ ].
**Note:**
Make sure to use ascending order for the interval and round to three decimal places as needed.
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