Confidence Intervals for a Proportion

Confidence Intervals for a Proportion

  • Sampling distribution of proportions is normal if both np and n(1-p) are >= 10.

  • Example: For p = 0.05 and n = 250,

    • Mean = 0.05,

    • Standard deviation (SE) = 0.014.

  • If np or n(1-p) < 10, the distribution is not normal (e.g., n = 50 → np = 2.5).

Central Limit Theorem (CLT)

  • CLT applies under certain conditions:

    • Independence of sampled observations.

    • Random sampling increases validity.

    • Assumption that sample mirrors population.

    • Sample size should ideally be > 30 for skewed distributions.

Parameter Estimation

  • Point estimates vary across samples; hence, margins of error provide a range.

  • Confidence intervals (CIs) give a plausible range for population parameters.

  • Example: 41% ± 2.9% → 95% confidence that the range is (38.1%, 43.9%).

Standard Error and Confidence Intervals

  • Sample proportions are normally distributed with mean = p and SE decreases as n increases.

  • Confidence interval formula: [ \hat{p} \pm z^* \times SE ]

  • Margin of error changes with confidence level adjusted by z value.

Unknown Population Proportion

  • Substitute ( \hat{p} ) for p in CLT conditions.

Confidence Interval Construction Steps

  1. Verify randomization and independence of data.

  2. Check if ( n\hat{p} ) and ( n(1 - \hat{p}) ) > 10.

  3. Calculate standard error (SE).

  4. Identify appropriate z-value for confidence level and calculate CI.

Interpretation of Confidence Intervals

  • e.g., 95% CI interpretation: "95% sure that X% to Y% of the population…"

  • Wider intervals increase confidence but may be less informative.

Z-Scores and Confidence Levels

  • Higher confidence levels correlate with wider intervals.

  • Common confidence levels: 90%, 95%, 98%, and 99%.

  • For 95% CI, use z = 1.96; for 98% CI typically z ≈ 2.33.

  • Margin of error affected by SE, confidence level, sample size but least by sample proportion.