ESTION 6 nufacturer needs to test to see the proportion of faulty products they produce. How large of a sample must the onfidence interval for this proportion with a width no greater than 14%

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### QUESTION 6

A manufacturer needs to test to see the proportion of faulty products they produce. How large of a sample must they use to get a 99% confidence interval for this proportion with a width no greater than 14%?

---
**Explanation:**

To determine the appropriate sample size for a confidence interval, consider the following steps:

1. **Identify the Desired Confidence Level:** Here, it is 99%.

2. **Determine the Maximum Acceptable Width of the Confidence Interval:** In this case, it is 14%.

3. **Estimate the Proportion of Faulty Products (p):**
   - Without a preliminary estimate, use p = 0.5 for the worst-case scenario, as it maximizes the sample size requirement.

4. **Use the Equation for Sample Size (n):**
   The formula for calculating the sample size for a proportion is:
   \[
   n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)
   \]
   Where:
   - \( n \) = required sample size
   - \( Z \) = Z-value (from the Z-table corresponding to the desired confidence level)
   - \( p \) = estimated proportion of faulty products
   - \( E \) = maximum acceptable error (half the width of the confidence interval)

5. **Find the Z-value for 99% Confidence:**
   - The Z-value for a 99% confidence level is approximately 2.576.

6. **Calculate the Required Sample Size:**
   - Assuming p = 0.5 and E = 0.07 (since 14% width means ±7%):
   \[
   n = \left( \frac{2.576^2 \cdot 0.5 \cdot (1 - 0.5)}{0.07^2} \right)
   \]
   - Simplify the calculation to find n.

This step-by-step approach will guide how to determine the necessary sample size to achieve the desired confidence interval.
Transcribed Image Text:### QUESTION 6 A manufacturer needs to test to see the proportion of faulty products they produce. How large of a sample must they use to get a 99% confidence interval for this proportion with a width no greater than 14%? --- **Explanation:** To determine the appropriate sample size for a confidence interval, consider the following steps: 1. **Identify the Desired Confidence Level:** Here, it is 99%. 2. **Determine the Maximum Acceptable Width of the Confidence Interval:** In this case, it is 14%. 3. **Estimate the Proportion of Faulty Products (p):** - Without a preliminary estimate, use p = 0.5 for the worst-case scenario, as it maximizes the sample size requirement. 4. **Use the Equation for Sample Size (n):** The formula for calculating the sample size for a proportion is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \] Where: - \( n \) = required sample size - \( Z \) = Z-value (from the Z-table corresponding to the desired confidence level) - \( p \) = estimated proportion of faulty products - \( E \) = maximum acceptable error (half the width of the confidence interval) 5. **Find the Z-value for 99% Confidence:** - The Z-value for a 99% confidence level is approximately 2.576. 6. **Calculate the Required Sample Size:** - Assuming p = 0.5 and E = 0.07 (since 14% width means ±7%): \[ n = \left( \frac{2.576^2 \cdot 0.5 \cdot (1 - 0.5)}{0.07^2} \right) \] - Simplify the calculation to find n. This step-by-step approach will guide how to determine the necessary sample size to achieve the desired confidence interval.
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