Assume that a sample is used to estimate a population proportion p. Find the 99.9% confidence interval for a sample of size 109 with 16 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places. 99.9% C.I. = Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.
Assume that a sample is used to estimate a population proportion p. Find the 99.9% confidence interval for a sample of size 109 with 16 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places. 99.9% C.I. = Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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![### Population Proportion Confidence Interval Calculation
In statistical analysis, determining the confidence interval for a population proportion is critical in understanding the accuracy of our sample estimates. Here, we will illustrate how to find the 99.9% confidence interval for a given sample.
#### Problem Statement
Assume that a sample is used to estimate a population proportion \( p \). We are given the following data:
- Sample size (\( n \)): 109
- Number of successes (\( x \)): 16
We need to find the 99.9% confidence interval for this sample and provide our answer in the form of an **open interval** (i.e., using parentheses), expressed as decimals (not percentages), accurate to three decimal places.
#### Steps to Calculate
1. **Calculate the Sample Proportion (\( \hat{p} \))**:
\[
\hat{p} = \frac{x}{n} = \frac{16}{109} \approx 0.147
\]
2. **Find the Standard Error (SE)**:
\[
SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} = \sqrt{\frac{0.147 \times (1 - 0.147)}{109}} \approx 0.034
\]
3. **Determine the Critical Value (z) for 99.9% Confidence**:
Using standard z-tables or statistical software, the critical value for a 99.9% confidence level is approximately 3.291.
4. **Compute the Margin of Error (ME)**:
\[
ME = z \times SE = 3.291 \times 0.034 \approx 0.112
\]
5. **Calculate the Confidence Interval**:
\[
\text{Lower Bound} = \hat{p} - ME \approx 0.147 - 0.112 = 0.035
\]
\[
\text{Upper Bound} = \hat{p} + ME \approx 0.147 + 0.112 = 0.259
\]
6. **Express the Interval**:
For our final answer, we express the interval as an open interval:
\[
99.9\% \text{ C.I.} = (0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa28b18f-3e2d-4592-9d09-7b963ee49630%2F9c0c407b-c690-4f05-a463-1c9d57e1848d%2F18ukyr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Population Proportion Confidence Interval Calculation
In statistical analysis, determining the confidence interval for a population proportion is critical in understanding the accuracy of our sample estimates. Here, we will illustrate how to find the 99.9% confidence interval for a given sample.
#### Problem Statement
Assume that a sample is used to estimate a population proportion \( p \). We are given the following data:
- Sample size (\( n \)): 109
- Number of successes (\( x \)): 16
We need to find the 99.9% confidence interval for this sample and provide our answer in the form of an **open interval** (i.e., using parentheses), expressed as decimals (not percentages), accurate to three decimal places.
#### Steps to Calculate
1. **Calculate the Sample Proportion (\( \hat{p} \))**:
\[
\hat{p} = \frac{x}{n} = \frac{16}{109} \approx 0.147
\]
2. **Find the Standard Error (SE)**:
\[
SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} = \sqrt{\frac{0.147 \times (1 - 0.147)}{109}} \approx 0.034
\]
3. **Determine the Critical Value (z) for 99.9% Confidence**:
Using standard z-tables or statistical software, the critical value for a 99.9% confidence level is approximately 3.291.
4. **Compute the Margin of Error (ME)**:
\[
ME = z \times SE = 3.291 \times 0.034 \approx 0.112
\]
5. **Calculate the Confidence Interval**:
\[
\text{Lower Bound} = \hat{p} - ME \approx 0.147 - 0.112 = 0.035
\]
\[
\text{Upper Bound} = \hat{p} + ME \approx 0.147 + 0.112 = 0.259
\]
6. **Express the Interval**:
For our final answer, we express the interval as an open interval:
\[
99.9\% \text{ C.I.} = (0
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