Notes on Confidence Intervals for Proportions

Module Overview

  • Module 5 focuses on Confidence Intervals for Proportions.
  • Builds on previous knowledge from earlier modules:
    • Modules 1-2: Data distributions, sampling, and experiments.
    • Module 3: Probability and random variables.
    • Module 4: Sampling distributions.
    • Module 5-6: Statistical inference concerning population parameters.

Sampling Distribution of Proportion

  • Definition: Population proportion, denoted as pp, is the ratio of successful outcomes (successes) in a population to the total population size.
  • Key Rules:
    • Rule 1: Mean of sampling distribution: μ(p^)=p\mu(\hat{p}) = p
    • Rule 2: Standard deviation: σp^=p(1−p)n\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}
    • Rule 3: Normal approximation valid when nn is large (usually np≥10np \geq 10 and n(1−p)≥10n(1-p) \geq 10).

Confidence Intervals for Proportions

  • Objective: Use sample data to infer results about a population.
  • Estimation Methods:
    • Point Estimation: Provides a single value as an estimate of a population parameter (e.g., sample mean yˉ\bar{y} for population mean μ\mu).
    • Interval Estimation: Offers a range of values likely to contain the population parameter based on sample data.

Understanding Interval Estimates

  • Point Estimates: Provide no measure of how close the estimate is to the true value.
  • Interval Estimates: Account for error/uncertainty, thus providing confidence intervals.

Standard Error

  • Standard error (SE) quantifies the uncertainty in estimates:
    • For proportions: SE(p^)=p^(1−p^)nSE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
    • For means: SE(yˉ)=σnSE(\bar{y}) = \frac{\sigma}{\sqrt{n}}

Confidence Level and Intervals

  • 95% Confidence Level: If 2 standard errors are considered, we are 95% confident the true proportion falls within this range.
  • Confidence Interpretation: When constructing many confidence intervals from different samples, approximately 95% will capture the true population parameter.

Margin of Error (M.E.)

  • Formula: M.E.=Point estimate±critical value×SE(estimate)M.E. = \text{Point estimate} \pm \text{critical value} \times SE(\text{estimate})
  • Trade-off between certainty and precision: Higher confidence requires a wider interval.

Critical Values

  • Determined using the 68-95-99.7% rule from the normal distribution:
    • Critical value (for 95% C.I.): approximately 1.96.
  • Table Overview: Approximate critical values for different confidence levels:
    • 90%: 1.645
    • 95%: 1.96
    • 99%: 2.576

Example Calculations

  1. Data Example:
    • 200 songs, 120 by male artists:
      • p^=120200=0.6\hat{p} = \frac{120}{200} = 0.6
    • Compute confidence intervals:
      • 90%: 0.6±1.6450.6(1−0.6)200=(0.54,0.66)0.6 \pm 1.645 \sqrt{\frac{0.6(1-0.6)}{200}} = (0.54, 0.66)
      • 95%: 0.6±1.960.6(1−0.6)200=(0.53,0.67)0.6 \pm 1.96 \sqrt{\frac{0.6(1-0.6)}{200}} = (0.53, 0.67)
      • 99%: 0.6±2.5760.6(1−0.6)200=(0.51,0.69)0.6 \pm 2.576 \sqrt{\frac{0.6(1-0.6)}{200}} = (0.51, 0.69)
  2. Understanding Interval Statements:
    • Avoid probabilistic phrasing. Instead focus on the process of construction.

Adjusting for Small Samples

  • Adjustment Method: When success/failure condition fails, add 2 successes and 2 failures to increase sample stability.
  • New estimate: p~=y+2n+4\tilde{p} = \frac{y + 2}{n + 4}

Sample Size Determination

  • Formula for sample size: n=(z∗×ME2×p^(1−p^))2n = \left( \frac{z^* \times ME}{2 \times \hat{p}(1 - \hat{p})} \right)^2
    • Use p^=0.5\hat{p} = 0.5 for conservative estimates if no prior information is available.

Conclusion

  • Key takeaway from this module is the understanding and ability to construct and interpret confidence intervals for population proportions based on sample data.