Day 21

Math 216: Statistical Thinking

Bastola

Confidence Intervals for Proportions: Statistical Inference Foundation

Key Question: How can we estimate population proportions with quantified uncertainty? Confidence intervals provide the mathematical framework for making reliable inferences about categorical data!

Real-World Applications:

  • Political Polling: Estimate voter support with margin of error
  • Market Research: Determine customer preference rates with confidence
  • Quality Control: Monitor defect rates in manufacturing processes
  • Medical Research: Track treatment success rates in clinical trials
  • Public Health: Estimate disease prevalence from survey samples

Statistical Framework: From Sample to Population

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
  • ⚖️ Confidence Interval: Range of plausible values for the true population proportion

Key Insight: Confidence intervals quantify the precision of our estimates and provide a range of plausible values for the true population proportion!

Visualizing Sampling Distributions for Proportions

Properties of Sampling Distribution for Proportions

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!

Large-Sample Confidence Interval for \(p\)

Confidence Interval Formula

General Formula: \[\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

Components:

  • \(\hat{p} = \frac{x}{n}\): Sample proportion (successes/total)
  • \(z_{\alpha/2}\): Critical value from standard normal distribution
  • \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\): Standard error of \(\hat{p}\)

Common Confidence Levels:

  • 90% confidence: \(z_{0.05} = 1.645\)
  • 95% confidence: \(z_{0.025} = 1.960\)
  • 99% confidence: \(z_{0.005} = 2.576\)

Key Insight: This formula provides a range of plausible values for the true population proportion!

Conditions for Valid Large-Sample C.I. for \(\boldsymbol{p}\)

Statistical Assumptions

Essential Conditions:

  1. Random Sample: Data collected through random sampling from target population
  2. Independence: Individual observations are independent of each other
  3. Success-Failure Condition: \(n\hat{p} \geq 15\) and \(n(1-\hat{p}) \geq 15\)

Verification Examples:

Example 1: Political Polling

  • Sample size: \(n = 1000\), observed support: \(\hat{p} = 0.52\)
  • \(n\hat{p} = 1000 \times 0.52 = 520 \geq 15\) ✅
  • \(n(1-\hat{p}) = 1000 \times 0.48 = 480 \geq 15\) ✅
  • Conclusion: Conditions satisfied, CI valid

Example 2: Market Research

  • Sample size: \(n = 100\), preference rate: \(\hat{p} = 0.30\)
  • \(n\hat{p} = 100 \times 0.30 = 30 \geq 15\) ✅
  • \(n(1-\hat{p}) = 100 \times 0.70 = 70 \geq 15\) ✅
  • Conclusion: Conditions satisfied, CI valid

Confidence Interval Visualization

Case Study 1: Political Polling Analysis

Context: National election polling with 1200 voters, observed support = 48%

Statistical Analysis:

We’re 95% confident the true support for Candidate A is between 45.2% and 50.8%

Case Study 2: Quality Control Analysis

Context: Manufacturing defect rate monitoring with 500 products, observed defect rate = 2%

Quality Control Analysis: We’re 95% confident the true defect rate is between 0.8% and 3.2%. If target defect rate is 1%, current process may need improvement

Case Study 3: Medical Research Analysis

Context: Clinical trial for new treatment with 800 patients, success rate = 65%

Medical Research Analysis:

We’re 95% confident the true treatment success rate is between 61.7% and 68.3%. The interval provides evidence for treatment effectiveness

Theoretical Formulas

Confidence Interval Calculations

Confidence Interval Formula: \[\text{CI} = \hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

Margin of Error: \[\text{ME} = z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

Sample Size Determination for Proportions

Planning Your Study

Key Question: How large a sample do we need to estimate a proportion with desired precision?

Sample Size Formula: \[n = \left(\frac{z_{\alpha/2}}{ME}\right)^2 \cdot p(1-p)\]

Components:

  • \(z_{\alpha/2}\): Critical value for desired confidence level
  • \(ME\): Desired margin of error
  • \(p\): Estimated proportion (use 0.5 for conservative approach)

Practical Application: This formula helps researchers plan studies with appropriate sample sizes to achieve desired precision!

Confidence Interval Exercises

Practical Exercises Using R

Exercise 1: Basic Confidence Interval Calculation

  • Sample data: 180 successes out of 400 trials

  • Calculate 95% CI:

    • \(\hat{p} = 180/400 = 0.45\)
    • \(SE = \sqrt{0.45 \times 0.55 / 400} = 0.0249\)
    • \(95\%\text{ CI} = 0.45 \pm 1.96 \times 0.0249 = (0.401, 0.499)\)

  • R code: prop.test(180, 400, conf.level = 0.95)$conf.int

Exercise 2: Sample Size Determination

  • Desired margin of error: 3%, estimated proportion: 50%, 95% confidence
  • Required sample size: \(n = \left(\frac{1.96}{0.03}\right)^2 \times 0.5 \times 0.5 = 1067.11 \rightarrow 1068\)
  • R code: ceiling((qnorm(0.975)/0.03)^2 * 0.5 * 0.5)

Conservative Sample Size Approach

When p is Unknown

The Conservative Approach: When we have no prior estimate of \(p\), use \(p = 0.5\)

Why 0.5?:

  • \(p(1-p)\) reaches its maximum value when \(p = 0.5\)
  • This gives the largest possible sample size
  • Guarantees desired margin of error regardless of true \(p\)

Conservative Sample Size Formula: \[n = \left(\frac{z_{\alpha/2}}{ME}\right)^2 \cdot 0.25\]

Example: For 95% confidence and 3% margin of error: \[n = \left(\frac{1.96}{0.03}\right)^2 \cdot 0.25 = 1067.11 \rightarrow 1068\]

Key Insight: Conservative approach ensures adequate sample size when prior information is unavailable!

Margin of Error vs Proportion Relationship

Sample Size Calculation Examples

Practical Applications

Example 1: Political Polling

  • Desired margin of error: 3%
  • Confidence level: 95%
  • Conservative approach (p unknown): \[n = \left(\frac{1.96}{0.03}\right)^2 \times 0.25 = 1067.11 \rightarrow 1068\]
  • With prior estimate (p = 0.52): \[n = \left(\frac{1.96}{0.03}\right)^2 \times 0.52 \times 0.48 = 1066.05 \rightarrow 1067\]

Example 2: Market Research

  • Desired margin of error: 5%
  • Confidence level: 90%
  • Conservative approach: \[n = \left(\frac{1.645}{0.05}\right)^2 \times 0.25 = 270.6 \rightarrow 271\]
  • With prior estimate (p = 0.30): \[n = \left(\frac{1.645}{0.05}\right)^2 \times 0.30 \times 0.70 = 227.4 \rightarrow 228\]

Key Insight: Using prior information can reduce required sample size significantly!

Sample Size Calculation Exercises

Practice Problems

Exercise 5: Basic Sample Size Calculation

  • Desired margin of error: 4%
  • Confidence level: 95%
  • Conservative approach (p unknown)
  • Calculate required sample size

Exercise 6: Sample Size with Prior Information

  • Desired margin of error: 3%
  • Confidence level: 95%
  • Prior estimate: p = 0.25
  • Calculate required sample size
  • Compare with conservative approach

Exercise 7: Impact of Confidence Level

  • Desired margin of error: 5%

  • Compare sample sizes for:

    • 90% confidence level
    • 95% confidence level
    • 99% confidence level

  • Use conservative approach

Exercise 8: Real-World Planning

  • Market research firm needs to estimate customer satisfaction
  • Desired margin of error: 2%
  • Confidence level: 95%
  • No prior information available
  • Calculate required sample size
  • If budget limits sample to 1500, what margin of error can be achieved?

Confidence Interval Exercises

Practical Exercises Using R

Exercise 3: Confidence Level Impact

  • Same data: 250 successes out of 500 trials
  • 90% CI: \(0.50 \pm 1.645 \times \sqrt{0.5 \times 0.5 / 500} = (0.463, 0.537)\)
  • 95% CI: \(0.50 \pm 1.96 \times \sqrt{0.5 \times 0.5 / 500} = (0.456, 0.544)\)
  • 99% CI: \(0.50 \pm 2.576 \times \sqrt{0.5 \times 0.5 / 500} = (0.442, 0.558)\)

Exercise 4: Real-World Interpretation

  • Market research: 60% prefer new product (95% CI: 55% to 65%, n=400)
  • Interpretation: We’re 95% confident the true preference rate is between 55% and 65%
  • Business decision: Strong evidence for product launch (interval well above 50%)

Connection with Previous Topics

Building on Statistical Foundations

Relationship to Sampling Distributions (Day 16-18):

  • Confidence intervals rely on the sampling distribution properties
  • Standard error quantifies the precision of our estimates
  • Central Limit Theorem ensures normal approximation for large samples

Relationship to Confidence Intervals for Means (Day 19-20):

  • Same conceptual framework: estimate ± margin of error
  • Different standard error formulas: \(\sigma/\sqrt{n}\) vs \(\sqrt{p(1-p)/n}\)
  • Same critical values from normal/t-distributions

Statistical Continuity:

  • All confidence intervals quantify uncertainty in parameter estimation
  • The interpretation remains consistent across different parameter types
  • The methodology builds systematically from basic principles

Key Statistical Principles

Essential Statistical Concepts:

  1. Uncertainty Quantification: Confidence intervals provide ranges of plausible values
  2. Precision Measurement: Standard error quantifies estimation accuracy
  3. Sample Size Effect: Larger samples → narrower intervals → more precise estimates
  4. Distribution Independence: CLT works with any population distribution

Statistical Guidelines:

  • Success-Failure Condition: Ensure \(n\hat{p} \geq 15\) and \(n(1-\hat{p}) \geq 15\)
  • Random Sampling: Essential for valid statistical inference
  • Confidence Level: Choose appropriate level for research context
  • Interpretation: Focus on the range, not just the point estimate

Next Topic: Applying these principles to hypothesis testing for proportions