Day 18

Math 216: Statistical Thinking

Bastola

Sampling Distributions for Proportions: Statistical Inference Foundation

Key Question: How can we accurately estimate population proportions from sample data? Sampling distributions provide the theoretical framework for statistical inference with categorical data!

Real-World Applications:

  • Political Polling: Estimate voter support from representative samples
  • Market Research: Understand customer preferences and behaviors
  • Quality Control: Monitor defect rates in manufacturing processes
  • Medical Research: Track treatment success rates in clinical trials

Statistical Framework: From Population to Sample

Statistical Framework:

  • 🎯 Population Proportion (p): True but unknown characteristic of entire population
  • 🔍 Sample Proportion (\(\hat{p}\)): Estimate calculated from sample data
  • 📊 Sampling Distribution: Pattern showing how \(\hat{p}\) varies across different samples
  • ⚖️ Standard Error: Quantifies precision of our estimate

Key Insight: Even though each sample gives a different answer, they follow a predictable normal pattern!

Visualizing Sampling Distributions

Properties of Sampling Distribution

Core Statistical Properties

Mean of \(\hat{p}\):

  • Unbiased Estimator: The expected value equals the population proportion
  • Mathematical Form: \(E(\hat{p}) = \mu_{\hat{p}} = p\)
  • Interpretation: On average, sample proportions equal the true population proportion

Standard Error of \(\hat{p}\):

  • Precision Measure: Quantifies variability across different samples
  • Mathematical Form: \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\)
  • Interpretation: Larger samples → smaller standard error → more precise estimates

Key Insight: These properties enable statistical inference from samples to populations!

Central Limit Theorem for Proportions

Core Statistical Principle

Central Limit Theorem for Proportions: For sufficiently large samples, the sampling distribution of \(\hat{p}\) is approximately normal, regardless of the population distribution shape.

Mathematical Formulation: \[\hat{p} \sim N\left(p, \sqrt{\frac{p(1-p)}{n}}\right)\]

Sample Size Requirements:

  • Success-Failure Condition: \(np \geq 15\) and \(n(1-p) \geq 15\)
  • Practical Interpretation: Ensure enough successes and failures for normal approximation

Statistical Significance: This universal principle enables confidence intervals and hypothesis testing for proportions!

CLT Condition Verification

Central Limit Theorem Conditions

Success-Failure Condition: For normal approximation to be valid, we need: \[np \geq 15 \quad \text{and} \quad n(1-p) \geq 15\]

Verification Examples:

Example 1: Political Polling

  • Population proportion: \(p = 0.52\) (52% support)
  • Sample size: \(n = 1000\)
  • \(np = 1000 \times 0.52 = 520 \geq 15\) ✅
  • \(n(1-p) = 1000 \times 0.48 = 480 \geq 15\) ✅
  • Conclusion: CLT applies, normal approximation valid

Example 2: Market Research

  • Population proportion: \(p = 0.30\) (30% preference)
  • Sample size: \(n = 100\)
  • \(np = 100 \times 0.30 = 30 \geq 15\) ✅
  • \(n(1-p) = 100 \times 0.70 = 70 \geq 15\) ✅
  • Conclusion: CLT applies, normal approximation valid

Z-Scores and Probability Calculations: Statistical Inference Tools

Z-Tables: Traditional Probability Calculation

Sampling Distribution Simulator

Probability Calculations

Probability Calculation Exercises

Real-World Application

  • Population proportion: \(p = 0.45\), sample size: \(n = 400\)

  • Standard error: \(SE = \sqrt{\frac{0.45 \times 0.55}{400}} = 0.0249\)

  • Find \(P(\hat{p} < 0.40)\):

    pnorm((0.40-0.45)/0.0249) = 0.0222

  • Find \(P(\hat{p} > 0.50)\):

    1 - pnorm((0.50-0.45)/0.0249) = 0.0222

  • Find \(P(0.42 < \hat{p} < 0.48)\):

    pnorm((0.48-0.45)/0.0249) - pnorm((0.42-0.45)/0.0249) = 0.7699