To find the 99.5% confidence interval for a population proportion $p$, we use the following formula for the confidence interval (CI):

$CI = \left( \hat{p} - Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \ \hat{p} + Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \right)$

Where:

• $\hat{p}$ is the sample proportion.
• $Z_{\alpha/2}$ is the critical value corresponding to the 99.5% confidence level.
• $n$ is the sample size.

Step 1: Identify the values

• Sample size $n = 329$
• Sample proportion $\hat{p} = 48.9\% = 0.489$
• Confidence level = 99.5%, which means $\alpha = 1 - 0.995 = 0.005$, so $\alpha/2 = 0.0025$.

The critical value $Z_{\alpha/2}$ for a 99.5% confidence interval (using standard normal distribution tables) is approximately 2.807.

Step 2: Calculate the margin of error

The margin of error (ME) is given by: $ME = Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

Substitute the values:

= 2.807 \times \sqrt{\frac{0.489 \times 0.511}{329}} = 2.807 \times \sqrt{\frac{0.249459}{329}} = 2.807 \times \sqrt{0.0007585} = 2.807 \times 0.02754$$ $$ME \approx 0.07734$$ ### Step 3: Find the confidence interval Now, calculate the confidence interval: $$CI = ( \hat{p} - ME, \ \hat{p} + ME ) = ( 0.489 - 0.07734, \ 0.489 + 0.07734 ) = ( 0.41166, \ 0.56634 )$$ ### Final Answer: The 99.5% confidence interval is: $$(0.412, 0.566)$$ Would you like more details or have any questions? Here are 5 related questions to extend this: 1. What is the interpretation of a 99.5% confidence interval in statistics? 2. How does increasing the confidence level affect the width of the confidence interval? 3. What happens if the sample size is increased to 500 with the same sample proportion? 4. How would the confidence interval change if we wanted a 95% confidence level instead? 5. What is the formula for finding the critical value $$Z_{\alpha/2}$$ from a standard normal distribution? **Tip:** The width of a confidence interval depends on both the sample size and the confidence level—the higher the confidence level, the wider the interval.