Calculate the confidence interval for a population proportion based on sample data.

## Understanding Confidence Intervals for Proportions

A confidence interval for a proportion estimates the range within which the true population proportion is likely to fall, based on sample data and the chosen confidence level.

### Formula for Confidence Interval of Proportion

The formula for calculating the confidence interval for a sample proportion \( \hat{p} \) is:

$$ CI = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} $$

where:

- \( \hat{p} \): Sample proportion
- \( n \): Sample size
- \( Z \): Z-score corresponding to the desired confidence level (e.g., 1.96 for 95%)

### Real-Life Example: Estimating Voting Preferences

Suppose a poll found that 52% of a sample of 1,000 voters support a particular candidate. To determine the 95% confidence interval for this proportion:

**Step 1:**Set \( \hat{p} = 0.52 \) and \( n = 1000 \).**Step 2:**Use a 95% confidence level, so \( Z = 1.96 \).**Step 3:**Calculate the standard error:**Step 4:**Calculate the margin of error:**Step 5:**Determine the confidence interval:

$$ SE = \sqrt{\frac{0.52 \times (1 - 0.52)}{1000}} \approx 0.0157 $$

$$ ME = 1.96 \times 0.0157 \approx 0.0308 $$

$$ CI = 0.52 \pm 0.0308 = [0.4892, 0.5508] $$

This means we can be 95% confident that between 48.92% and 55.08% of the population supports the candidate.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.