In statistical analysis, estimating and interpreting confidence intervals for population proportions is a fundamental concept. This section explores how to construct these intervals using large sample data, emphasizing the necessary approximations and formulas.

## Confidence Intervals

### Key Concepts:

**Population Proportion**$p$: True proportion of a characteristic in the whole population.**Sample Proportion**$(\hat{p})$: Proportion observed in a sample, used to estimate the population proportion.**Confidence Level:**The probability (usually expressed as a percentage) that the confidence interval includes the true population proportion.

## Calculating Confidence Intervals

**General Formula:**$\hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$- $\hat{p}$ is the sample proportion, $Z$ is the Z-score for the confidence level, and $(n)$ is the sample size.
**Z-scores:**1.645 (90%), 1.96 (95%), 2.576 (99%).**Sample Size Requirement:**$n\hat{p}$ and $n(1 - \hat{p})$ should be greater than 5 for a normal approximation.

## Problem Examples

### Example 1: Constructing a Confidence Interval

A survey of 400 people where 220 prefer a specific brand. Find the 95% confidence interval for the population's preference proportion.

**Solution:**

**Sample Proportion**$(\hat{p})$: $\hat{p} = \frac{220}{400} = 0.55$.**Sample Size (n)**: $n = 400$.**Z-Score for 95% Confidence**: $Z = 1.96$.**Confidence Interval Formula**: $\hat{p} \pm Z \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}$.**Margin of Error**: $\approx 0.048.$**Confidence Interval**: $[0.502, 0.598]$.

**Conclusion**: There's a 95% confidence that between 50.2% and 59.8% of the population prefers the brand.

### Example 2: Interpreting a Confidence Interval

**Scenario:**A 95% confidence interval is [0.48, 0.62].**Interpretation:**- We are 95% confident the true population proportion is between 48% and 62%.
- "95% Confidence" means about 95% of such intervals from repeated samples will contain the true proportion.
- A wider interval suggests more uncertainty, a narrower interval suggests greater precision.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.