Calculate the confidence interval for the difference in proportions between two populations.

## Understanding Confidence Intervals for Difference in Proportions

A confidence interval for the difference in proportions estimates the range within which the true difference between two population proportions is likely to fall, based on sample data and the chosen confidence level.

### Formula for Confidence Interval of Difference in Proportions

The formula to calculate the confidence interval for the difference in proportions \( \hat{p}_1 \) and \( \hat{p}_2 \) is:

$$ CI = (\hat{p}_1 - \hat{p}_2) \pm Z \cdot \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}} $$

where:

- \( \hat{p}_1 \): Proportion of success in the first sample
- \( \hat{p}_2 \): Proportion of success in the second sample
- \( n_1 \): Sample size of the first group
- \( n_2 \): Sample size of the second group
- \( Z \): Z-score associated with the desired confidence level (e.g., 1.96 for 95%)

### Real-Life Example: Comparing Customer Satisfaction

Suppose a company wants to compare customer satisfaction between two products. They survey customers and find that 60% of 150 customers liked Product A, while 50% of 200 customers liked Product B. We can calculate the 95% confidence interval for the difference in proportions as follows:

**Step 1:**Set \( \hat{p}_1 = 0.6 \), \( n_1 = 150 \), \( \hat{p}_2 = 0.5 \), and \( n_2 = 200 \).**Step 2:**Use the 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:

$$ SE = \sqrt{\frac{0.6 \times (1 - 0.6)}{150} + \frac{0.5 \times (1 - 0.5)}{200}} \approx 0.0578 $$

$$ ME = 1.96 \times 0.0578 \approx 0.1133 $$

$$ CI = (0.6 - 0.5) \pm 0.1133 = 0.1 \pm 0.1133 $$

Thus, the 95% confidence interval for the difference in proportions is approximately \([ -0.0133, 0.2133 ]\).

This means the company can be 95% confident that the true difference in customer satisfaction between the two products falls within the range of -0.0133 to 0.2133.

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.