Express the confidence interval (0.069,0.161) in the form of p-E
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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![### Expressing the Confidence Interval
To express the confidence interval \((0.069, 0.161)\) in the form of \(\hat{p} - E < p < \hat{p} + E\), follow the steps below:
1. **Identify the midpoint (or point estimate \(\hat{p}\)):**
- The midpoint \(\hat{p}\) is the average of the two endpoints of the interval:
\[
\hat{p} = \frac{0.069 + 0.161}{2}
\]
2. **Calculate \(\hat{p}\):**
\[
\hat{p} = \frac{0.069 + 0.161}{2} = \frac{0.23}{2} = 0.115
\]
3. **Determine the margin of error (\(E\)):**
- The margin of error \(E\) is the distance from the midpoint \(\hat{p}\) to either endpoint:
\[
E = \hat{p} - 0.069 = 0.115 - 0.069 = 0.046
\]
- Alternatively:
\[
E = 0.161 - \hat{p} = 0.161 - 0.115 = 0.046
\]
Thus, the confidence interval \((0.069, 0.161)\) can be expressed in the form \(\hat{p} - E < p < \hat{p} + E\) as:
\(\boxed{0.069 < p < 0.161}\)
In conclusion, we have:
\[
0.069 < p < 0.161
\]
(Type integers or decimals.)
This completes the transcribed explanation for expressing the given confidence interval in the required mathematical form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86445a01-0a3f-49cc-9d22-b352e3028fb5%2F0c882895-3194-41f8-b88a-6ab328e5cb22%2F74g1lse_processed.png&w=3840&q=75)
Transcribed Image Text:### Expressing the Confidence Interval
To express the confidence interval \((0.069, 0.161)\) in the form of \(\hat{p} - E < p < \hat{p} + E\), follow the steps below:
1. **Identify the midpoint (or point estimate \(\hat{p}\)):**
- The midpoint \(\hat{p}\) is the average of the two endpoints of the interval:
\[
\hat{p} = \frac{0.069 + 0.161}{2}
\]
2. **Calculate \(\hat{p}\):**
\[
\hat{p} = \frac{0.069 + 0.161}{2} = \frac{0.23}{2} = 0.115
\]
3. **Determine the margin of error (\(E\)):**
- The margin of error \(E\) is the distance from the midpoint \(\hat{p}\) to either endpoint:
\[
E = \hat{p} - 0.069 = 0.115 - 0.069 = 0.046
\]
- Alternatively:
\[
E = 0.161 - \hat{p} = 0.161 - 0.115 = 0.046
\]
Thus, the confidence interval \((0.069, 0.161)\) can be expressed in the form \(\hat{p} - E < p < \hat{p} + E\) as:
\(\boxed{0.069 < p < 0.161}\)
In conclusion, we have:
\[
0.069 < p < 0.161
\]
(Type integers or decimals.)
This completes the transcribed explanation for expressing the given confidence interval in the required mathematical form.
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