Express the confidence interval 0.333
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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**Express the Confidence Interval**
Problem Statement:
Express the confidence interval \(0.333 < p < 0.777\) in the form \(\hat{p} \pm E\).
Expression:
\[ \hat{p} \pm E = \boxed{} \pm \boxed{} \]
---
Here, the image provides a task in mathematical statistics. The problem requires expressing the given confidence interval in a specific statistical notation used for probabilities.
To solve this, recall that the confidence interval bounds are given by:
\[ p = \hat{p} - E \quad \text{to} \quad p = \hat{p} + E, \]
where \(\hat{p}\) is the estimated proportion and \(E\) is the margin of error.
Given the interval \(0.333 < p < 0.777\):
1. The midpoint \(\hat{p}\) (the point estimate) can be calculated as:
\[ \hat{p} = \frac{0.333 + 0.777}{2} = 0.555. \]
2. The margin of error \(E\) is the distance from the midpoint to either endpoint:
\[ E = 0.777 - 0.555 = 0.222. \]
Thus, the confidence interval in the form \(\hat{p} \pm E\) is:
\[ \hat{p} \pm E = 0.555 \pm 0.222. \]
Accordingly, the final answer should fill in the boxes provided in the image:
\[ \hat{p} \pm E = \boxed{0.555} \pm \boxed{0.222}. \]
This form clearly communicates the estimated proportion and the margin of error for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6eeb107-3976-477c-976e-1778ebbdfd8b%2F5c8c3d38-ca50-40a2-99b7-273bbd78355e%2F23gi796_processed.png&w=3840&q=75)
Transcribed Image Text:---
**Express the Confidence Interval**
Problem Statement:
Express the confidence interval \(0.333 < p < 0.777\) in the form \(\hat{p} \pm E\).
Expression:
\[ \hat{p} \pm E = \boxed{} \pm \boxed{} \]
---
Here, the image provides a task in mathematical statistics. The problem requires expressing the given confidence interval in a specific statistical notation used for probabilities.
To solve this, recall that the confidence interval bounds are given by:
\[ p = \hat{p} - E \quad \text{to} \quad p = \hat{p} + E, \]
where \(\hat{p}\) is the estimated proportion and \(E\) is the margin of error.
Given the interval \(0.333 < p < 0.777\):
1. The midpoint \(\hat{p}\) (the point estimate) can be calculated as:
\[ \hat{p} = \frac{0.333 + 0.777}{2} = 0.555. \]
2. The margin of error \(E\) is the distance from the midpoint to either endpoint:
\[ E = 0.777 - 0.555 = 0.222. \]
Thus, the confidence interval in the form \(\hat{p} \pm E\) is:
\[ \hat{p} \pm E = 0.555 \pm 0.222. \]
Accordingly, the final answer should fill in the boxes provided in the image:
\[ \hat{p} \pm E = \boxed{0.555} \pm \boxed{0.222}. \]
This form clearly communicates the estimated proportion and the margin of error for educational purposes.
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