Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. They randomly survey 381 drivers and find that 320 claim to always buckle up. Construct a 95% confidence interval for the population proportion that claim to always buckle up.
Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car. They randomly survey 381 drivers and find that 320 claim to always buckle up. Construct a 95% confidence interval for the population proportion that claim to always buckle up.
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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![**Educational Content: Understanding Confidence Intervals**
Insurance companies are interested in knowing the percentage of drivers who always buckle up before riding in a car. They randomly survey 381 drivers and find that 320 claim to always buckle up.
**Objective:**
Construct a 95% confidence interval for the population proportion that claims to always buckle up.
**Instructions:**
- Use the sample data to determine the population proportion confidence interval.
- Do not round intermediate steps; round the final answer to four decimal places.
**Step-by-Step Guide:**
1. **Identify the Sample Proportion (\(\hat{p}\)):**
- Calculate \( \hat{p} = \frac{320}{381} \).
2. **Determine the Standard Error (SE):**
- Use the formula for the standard error of a proportion:
\[
\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]
3. **Find the Z-score for 95% Confidence Level:**
- Typically, the Z-score is 1.96 for a 95% confidence interval.
4. **Calculate the Confidence Interval:**
- Use the formula:
\[
\hat{p} \pm Z \times \text{SE}
\]
5. **Conclusion:**
- Interpreting the results, the confidence interval provides a range that is likely to contain the true population proportion of drivers who always buckle up with a 95% level of certainty.
This process helps understand how statistical confidence intervals are used to make inferences about a population based on a sample.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88a14f8c-9619-4f58-97b4-5c7c439fe2b7%2Fe8939246-b51f-4d23-801b-81f9fa9d3fb7%2Ffnfzoba_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Understanding Confidence Intervals**
Insurance companies are interested in knowing the percentage of drivers who always buckle up before riding in a car. They randomly survey 381 drivers and find that 320 claim to always buckle up.
**Objective:**
Construct a 95% confidence interval for the population proportion that claims to always buckle up.
**Instructions:**
- Use the sample data to determine the population proportion confidence interval.
- Do not round intermediate steps; round the final answer to four decimal places.
**Step-by-Step Guide:**
1. **Identify the Sample Proportion (\(\hat{p}\)):**
- Calculate \( \hat{p} = \frac{320}{381} \).
2. **Determine the Standard Error (SE):**
- Use the formula for the standard error of a proportion:
\[
\text{SE} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]
3. **Find the Z-score for 95% Confidence Level:**
- Typically, the Z-score is 1.96 for a 95% confidence interval.
4. **Calculate the Confidence Interval:**
- Use the formula:
\[
\hat{p} \pm Z \times \text{SE}
\]
5. **Conclusion:**
- Interpreting the results, the confidence interval provides a range that is likely to contain the true population proportion of drivers who always buckle up with a 95% level of certainty.
This process helps understand how statistical confidence intervals are used to make inferences about a population based on a sample.
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