To calculate a 95% confidence interval for the proportion of people who performed volunteer work, we can use the formula for a confidence interval for a proportion:

$\hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

where:

• $\hat{p} = \frac{x}{n}$ is the sample proportion,
• $x$ is the number of successes (people who did volunteer work),
• $n$ is the sample size,
• $Z$ is the Z-score corresponding to the desired confidence level.

Step 1: Calculate $\hat{p}$

Given:

• $x = 530$
• $n = 1297$

$\hat{p} = \frac{530}{1297} \approx 0.409$

Step 2: Find the Z-score

For a 95% confidence interval, the Z-score is approximately 1.96.

Step 3: Calculate the standard error (SE)

$SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.409(1 - 0.409)}{1297}} \approx \sqrt{\frac{0.409 \times 0.591}{1297}} \approx \sqrt{\frac{0.241}{1297}} \approx 0.014$

Step 4: Calculate the margin of error (ME)

$ME = Z \times SE = 1.96 \times 0.014 \approx 0.0274$

Step 5: Construct the confidence interval

$\text{Lower limit} = \hat{p} - ME \approx 0.409 - 0.0274 \approx 0.3816$ $\text{Upper limit} = \hat{p} + ME \approx 0.409 + 0.0274 \approx 0.4364$

Thus, the 95% confidence interval for the proportion of people who performed volunteer work during the past year is approximately:

$(0.382, 0.436)$

Summary

The 95% confidence interval for the proportion of people who performed volunteer work is approximately (0.382, 0.436).

Do you want details on any of the steps, or do you have any questions? Here are some follow-up questions you might consider:

1. What does the confidence interval indicate about the population proportion?
2. How does changing the confidence level affect the width of the interval?
3. What are some potential sources of error in this survey?
4. How would you calculate a confidence interval for a different sample size?
5. What assumptions must be met to use this method for calculating the confidence interval?

Tip: Always ensure the sample size is sufficiently large for the normal approximation to be valid when calculating confidence intervals for proportions.