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CIE A-Level Maths Study Notes

5.4.7 Confidence Interval for Population Proportion

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 pp: True proportion of a characteristic in the whole population.
  • Sample Proportion (p^)(\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: p^±Z×p^(1p^)n\hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
  • p^\hat{p} is the sample proportion, ZZ is the Z-score for the confidence level, and (n)(n) is the sample size.
  • Z-scores: 1.645 (90%), 1.96 (95%), 2.576 (99%).
  • Sample Size Requirement: np^n\hat{p} and n(1p^)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 (p^)(\hat{p}): p^=220400=0.55\hat{p} = \frac{220}{400} = 0.55.
  • Sample Size (n): n=400n = 400.
  • Z-Score for 95% Confidence: Z=1.96Z = 1.96.
  • Confidence Interval Formula: p^±Z×p^×(1p^)n\hat{p} \pm Z \times \sqrt{\frac{\hat{p} \times (1 - \hat{p})}{n}}.
  • Margin of Error: 0.048.\approx 0.048.
  • Confidence Interval: [0.502,0.598][0.502, 0.598].

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

Constructing a Confidence Interval Illustration

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.
Interpreting a Confidence Interval Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

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.

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