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Confidence Intervals

Confidence Intervals & Interval Estimation

Master the theory and practice of interval estimation: from fundamental concepts of coverage probability to advanced construction methods using pivotal quantities, with comprehensive applications to normal populations and beyond

10 Lessons6-8 HoursIntermediate Level
Learning Objectives
Master these key concepts and skills in confidence interval theory and construction
  • Understand interval estimation fundamentals and coverage probability concepts
  • Master confidence levels, confidence coefficients, and their proper interpretation
  • Learn pivotal quantity method for constructing confidence intervals
  • Apply confidence interval methods to normal population parameters
  • Construct two-sample confidence intervals for mean differences and variance ratios
  • Understand large sample approximations and Bootstrap confidence interval methods

Core Topics

Six fundamental areas covering the complete theory of confidence intervals and interval estimation

Interval Estimation Fundamentals
Core concepts of interval estimation, coverage probability, and evaluation criteria

Key Concepts:

  • Definition: Interval estimators [θ̂_L, θ̂_U] for parameter θ
  • Coverage probability: P_θ{θ ∈ [θ̂_L, θ̂_U]} measures reliability
  • Confidence coefficient: infimum of coverage probability over parameter space
  • Precision vs reliability tradeoff: shorter intervals vs higher confidence
Confidence Levels & Interpretation
Understanding confidence levels, equal-tailed vs one-sided intervals, and proper interpretation

Key Concepts:

  • Confidence level 1-α: P_θ{θ ∈ [θ̂_L, θ̂_U]} ≥ 1-α for all θ
  • Frequentist interpretation: ~100(1-α)% of intervals contain true θ
  • One-sided confidence bounds: upper and lower confidence limits
  • Equal confidence intervals: confidence coefficient exactly equals 1-α
Pivotal Quantity Method
The fundamental method for constructing confidence intervals using pivotal quantities

Key Concepts:

  • Definition: G(X̃, θ) with known distribution independent of unknown parameters
  • Construction steps: Find pivot → Determine quantiles → Algebraic manipulation
  • Examples: (X̄ - μ)/(σ/√n) ~ N(0,1) for normal populations
  • Extension to asymmetric distributions using distribution function monotonicity
Normal Population Confidence Intervals
Comprehensive coverage of confidence intervals for normal population parameters

Key Concepts:

  • Mean μ (σ² known): X̄ ± u_{α/2} · σ/√n using standard normal quantiles
  • Mean μ (σ² unknown): X̄ ± t_{α/2}(n-1) · S/√n using t-distribution
  • Variance σ² (μ known/unknown): (n-1)S²/χ²_{α/2}(n-1) bounds using chi-square
  • Sample size effects and practical considerations
Two-Sample Confidence Intervals
Confidence intervals for comparing parameters between two normal populations

Key Concepts:

  • Mean difference δ = μ₂ - μ₁ with known/unknown variances
  • Pooled variance approach when σ₁² = σ₂² (unknown)
  • Variance ratio γ = σ₁²/σ₂² using F-distribution
  • Large sample approximations and Welch's t-test
Non-Normal & Large Sample Methods
Extensions to non-normal populations and advanced interval construction methods

Key Concepts:

  • Central Limit Theorem applications for large samples
  • Bootstrap confidence intervals for unknown distributions
  • Discrete distributions: Poisson, binomial exact intervals
  • Wilson intervals for binomial proportions and continuity corrections

Theoretical Foundations

Mathematical foundations underlying confidence interval theory and construction

Coverage Probability Theory

The fundamental measure P_θ{θ ∈ [θ̂_L, θ̂_U]} quantifies how often intervals contain the true parameter. Unlike point estimation, θ is fixed and intervals are random.

P_θ{θ̂_L ≤ θ ≤ θ̂_U} = ∫ I{θ̂_L(x) ≤ θ ≤ θ̂_U(x)} f(x|θ) dx

Neyman Construction Principle

Minimize expected interval length subject to maintaining confidence level 1-α. This optimality criterion balances precision with reliability.

min E_θ[θ̂_U - θ̂_L] subject to inf_θ P_θ{θ ∈ [θ̂_L, θ̂_U]} ≥ 1-α

Pivotal Distribution Theory

A pivotal quantity G(X̃, θ) has distribution independent of unknown parameters, enabling direct probability statements for interval construction.

P{c ≤ G(X̃, θ) ≤ d} = 1-α ⟹ P{g₁(X̃) ≤ θ ≤ g₂(X̃)} = 1-α

Practical Examples

Real-world applications demonstrating confidence interval construction methods

🌱
Plant Height Measurement
Normal population mean confidence interval with known variance

Scenario:

Heights X ~ N(μ, 16), n = 36, x̄ = 15 cm, α = 0.05

Solution:

95% CI: 15 ± 1.96 × 4/6 = [13.693, 16.307] cm

Method:

Standard normal pivot with known σ = 4

🏭
Quality Control Variance
Normal population variance confidence interval

Scenario:

Measurements n = 4, x̄ = 8.4%, s = 0.03%, α = 0.05

Solution:

95% CI for σ²: [0.00029, 0.0125] using chi-square distribution

Method:

Chi-square pivot (n-1)S²/σ² ~ χ²(3)

💊
Treatment Comparison
Two-sample mean difference with equal unknown variances

Scenario:

Treatment vs Control, pooled variance approach

Solution:

Difference CI using t-distribution with pooled standard error

Method:

Pooled t-test with combined variance estimate

🎯 Apply Your Knowledge

Practice confidence interval construction with our interactive tools and comprehensive problem sets

Confidence Interval Calculator

Calculate confidence intervals for normal population means with interactive step-by-step solutions.

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Practice Problems

Test your understanding with problems covering pivotal quantities, normal intervals, and two-sample cases.

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Formula Reference

Quick reference for all confidence interval formulas, critical values, and construction methods.

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