Construct the 90% confidence interval estimate of the population proportion p if the sample size is n = 200 and the number of successes in the sample is = : 152.
Construct the 90% confidence interval estimate of the population proportion p if the sample size is n = 200 and the number of successes in the sample is = : 152.
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
![**Problem Statement:**
Construct the 90% confidence interval estimate of the population proportion \( p \) if the sample size is \( n = 200 \) and the number of successes in the sample is \( x = 152 \).
\[
\boxed{} < p < \boxed{}
\]
**Explanation:**
To construct the 90% confidence interval for the population proportion \( p \), you can use the formula for the confidence interval of a proportion:
1. Calculate the sample proportion (\( \hat{p} \)):
\[
\hat{p} = \frac{x}{n} = \frac{152}{200} = 0.76
\]
2. Find the standard error (SE) of the sample proportion:
\[
\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.76 \times 0.24}{200}}
\]
3. Use the Z-score for a 90% confidence level (approximately 1.645 for two-tailed tests).
4. Calculate the confidence interval:
\[
\hat{p} \pm Z \times \text{SE}
\]
5. Substitute the known values into the equation to find the interval.
The empty boxes represent the bounds of the confidence interval for \( p \) and should be filled with calculated values once the above steps are completed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20960b1b-ef33-4f67-8b23-d0d3dcfa80e2%2F29801d2a-4b2d-4436-a09e-bc6c48071828%2Fm2zrzpi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Construct the 90% confidence interval estimate of the population proportion \( p \) if the sample size is \( n = 200 \) and the number of successes in the sample is \( x = 152 \).
\[
\boxed{} < p < \boxed{}
\]
**Explanation:**
To construct the 90% confidence interval for the population proportion \( p \), you can use the formula for the confidence interval of a proportion:
1. Calculate the sample proportion (\( \hat{p} \)):
\[
\hat{p} = \frac{x}{n} = \frac{152}{200} = 0.76
\]
2. Find the standard error (SE) of the sample proportion:
\[
\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.76 \times 0.24}{200}}
\]
3. Use the Z-score for a 90% confidence level (approximately 1.645 for two-tailed tests).
4. Calculate the confidence interval:
\[
\hat{p} \pm Z \times \text{SE}
\]
5. Substitute the known values into the equation to find the interval.
The empty boxes represent the bounds of the confidence interval for \( p \) and should be filled with calculated values once the above steps are completed.
Expert Solution
Step 1: Given information
n=200, x=152
Step by step
Solved in 3 steps with 1 images
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