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.