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Formula Reference

Confidence Intervals Formulas

Comprehensive reference for confidence interval construction formulas, covering normal populations, two-sample inference, large sample methods, and specialized techniques

Interval Estimation Fundamentals
Basic Definitions
Core definitions and concepts in interval estimation theory

Key Formulas:

Interval Estimator: [θ^L,θ^U]\text{Interval Estimator: } [\hat{\theta}_L, \hat{\theta}_U]
Coverage Probability: Pθ{θ∈[θ^L,θ^U]}\text{Coverage Probability: } P_{\theta}\{\theta \in [\hat{\theta}_L, \hat{\theta}_U]\}
Confidence Coefficient: inf⁡θ∈ΘPθ{θ∈[θ^L,θ^U]}\text{Confidence Coefficient: } \inf_{\theta \in \Theta} P_{\theta}\{\theta \in [\hat{\theta}_L, \hat{\theta}_U]\}
Confidence Level: Pθ{θ∈[θ^L,θ^U]}≥1−α\text{Confidence Level: } P_{\theta}\{\theta \in [\hat{\theta}_L, \hat{\theta}_U]\} \geq 1-\alpha

Applications:

  • Parameter estimation with uncertainty quantification
  • Statistical inference with controlled reliability
  • Evaluation of estimator precision vs confidence trade-off
Pivotal Quantity Method
Construction Methods
The fundamental method for constructing confidence intervals using pivotal quantities

Key Formulas:

G(X~,θ) with distribution independent of unknown parametersG(\tilde{X}, \theta) \text{ with distribution independent of unknown parameters}
P{c≤G(X~,θ)≤d}=1−αP\{c \leq G(\tilde{X}, \theta) \leq d\} = 1-\alpha
c≤G(X~,θ)≤d⇒θ^L≤θ≤θ^Uc \leq G(\tilde{X}, \theta) \leq d \Rightarrow \hat{\theta}_L \leq \theta \leq \hat{\theta}_U
P{θ^L≤θ≤θ^U}=1−αP\{\hat{\theta}_L \leq \theta \leq \hat{\theta}_U\} = 1-\alpha

Applications:

  • Normal population parameter intervals
  • Exact confidence intervals for parametric families
  • Optimal interval construction under symmetry
Normal Population Mean (σ² Known)
Normal Intervals
Confidence intervals for normal population mean when variance is known

Key Formulas:

G=Xˉ−μσ/n∼N(0,1)G = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)
[Xˉ−uα/2⋅σn,Xˉ+uα/2⋅σn]\left[\bar{X} - u_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \bar{X} + u_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\right]
uα/2=Φ−1(1−α/2)u_{\alpha/2} = \Phi^{-1}(1-\alpha/2)
Margin of Error: uα/2⋅σn\text{Margin of Error: } u_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

Applications:

  • Quality control with known process variation
  • Measurement studies with established precision
  • Large sample approximations
Normal Population Mean (σ² Unknown)
Normal Intervals
Confidence intervals for normal population mean when variance is unknown

Key Formulas:

T=Xˉ−μS/n∼t(n−1)T = \frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n-1)
[Xˉ−tα/2(n−1)⋅Sn,Xˉ+tα/2(n−1)⋅Sn]\left[\bar{X} - t_{\alpha/2}(n-1) \cdot \frac{S}{\sqrt{n}}, \bar{X} + t_{\alpha/2}(n-1) \cdot \frac{S}{\sqrt{n}}\right]
S2=1n−1∑i=1n(Xi−Xˉ)2S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2
Margin of Error: tα/2(n−1)⋅Sn\text{Margin of Error: } t_{\alpha/2}(n-1) \cdot \frac{S}{\sqrt{n}}

Applications:

  • Small sample inference
  • Most common practical scenario
  • Robust to mild non-normality
Normal Population Variance (μ Known)
Normal Intervals
Confidence intervals for normal population variance when mean is known

Key Formulas:

G=∑i=1n(Xi−μ)2σ2∼χ2(n)G = \frac{\sum_{i=1}^n (X_i - \mu)^2}{\sigma^2} \sim \chi^2(n)
[∑i=1n(Xi−μ)2χα/22(n),∑i=1n(Xi−μ)2χ1−α/22(n)]\left[\frac{\sum_{i=1}^n (X_i - \mu)^2}{\chi^2_{\alpha/2}(n)}, \frac{\sum_{i=1}^n (X_i - \mu)^2}{\chi^2_{1-\alpha/2}(n)}\right]
Q2=∑i=1n(Xi−μ)2Q^2 = \sum_{i=1}^n (X_i - \mu)^2
χα/22(n),χ1−α/22(n)\chi^2_{\alpha/2}(n), \chi^2_{1-\alpha/2}(n)

Applications:

  • Process variation assessment
  • Quality control variance monitoring
  • Theoretical variance bounds
Normal Population Variance (μ Unknown)
Normal Intervals
Confidence intervals for normal population variance when mean is unknown

Key Formulas:

G=(n−1)S2σ2∼χ2(n−1)G = \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)
[(n−1)S2χα/22(n−1),(n−1)S2χ1−α/22(n−1)]\left[\frac{(n-1)S^2}{\chi^2_{\alpha/2}(n-1)}, \frac{(n-1)S^2}{\chi^2_{1-\alpha/2}(n-1)}\right]
S2=1n−1∑i=1n(Xi−Xˉ)2S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2
Degrees of Freedom: n−1\text{Degrees of Freedom: } n-1

Applications:

  • Most common variance estimation scenario
  • Manufacturing process control
  • Measurement precision assessment
Two Normal Means (σ₁², σ₂² Known)
Two-Sample Inference
Confidence intervals for the difference between two normal population means with known variances

Key Formulas:

δ=μ2−μ1\delta = \mu_2 - \mu_1
G=(Yˉ−Xˉ)−δσ12/m+σ22/n∼N(0,1)G = \frac{(\bar{Y} - \bar{X}) - \delta}{\sqrt{\sigma_1^2/m + \sigma_2^2/n}} \sim N(0,1)
(Yˉ−Xˉ)±uα/2σ12m+σ22n(\bar{Y} - \bar{X}) \pm u_{\alpha/2}\sqrt{\frac{\sigma_1^2}{m} + \frac{\sigma_2^2}{n}}
Standard Error: σ12m+σ22n\text{Standard Error: } \sqrt{\frac{\sigma_1^2}{m} + \frac{\sigma_2^2}{n}}

Applications:

  • Comparing treatment effects
  • A/B testing with known variability
  • Quality comparison studies
Two Normal Means (Equal Unknown Variances)
Two-Sample Inference
Confidence intervals for mean difference when populations have equal unknown variances

Key Formulas:

SW2=(m−1)SX2+(n−1)SY2m+n−2S_W^2 = \frac{(m-1)S_X^2 + (n-1)S_Y^2}{m+n-2}
T=(Yˉ−Xˉ)−δSW1/m+1/n∼t(m+n−2)T = \frac{(\bar{Y} - \bar{X}) - \delta}{S_W\sqrt{1/m + 1/n}} \sim t(m+n-2)
(Yˉ−Xˉ)±tα/2(m+n−2)⋅SW1m+1n(\bar{Y} - \bar{X}) \pm t_{\alpha/2}(m+n-2) \cdot S_W\sqrt{\frac{1}{m} + \frac{1}{n}}
σ12=σ22=σ2 (unknown)\sigma_1^2 = \sigma_2^2 = \sigma^2 \text{ (unknown)}

Applications:

  • Classical two-sample t-test
  • Treatment vs control comparisons
  • Most common two-sample scenario
Two Normal Variance Ratio
Two-Sample Inference
Confidence intervals for the ratio of two normal population variances

Key Formulas:

γ=σ12σ22\gamma = \frac{\sigma_1^2}{\sigma_2^2}
F=SX2/SY2σ12/σ22∼F(m−1,n−1)F = \frac{S_X^2/S_Y^2}{\sigma_1^2/\sigma_2^2} \sim F(m-1, n-1)
[SX2SY2⋅1Fα/2(m−1,n−1),SX2SY2⋅1F1−α/2(m−1,n−1)]\left[\frac{S_X^2}{S_Y^2} \cdot \frac{1}{F_{\alpha/2}(m-1,n-1)}, \frac{S_X^2}{S_Y^2} \cdot \frac{1}{F_{1-\alpha/2}(m-1,n-1)}\right]
F1−α/2(a,b)=1Fα/2(b,a)F_{1-\alpha/2}(a,b) = \frac{1}{F_{\alpha/2}(b,a)}

Applications:

  • Testing equality of variances
  • Comparing process variabilities
  • Validating equal variance assumptions
Large Sample Approximations
Asymptotic Methods
Confidence intervals using Central Limit Theorem for large samples

Key Formulas:

Xˉ→DN(μ,σ2n) as n→∞\bar{X} \stackrel{D}{\to} N\left(\mu, \frac{\sigma^2}{n}\right) \text{ as } n \to \infty
Xˉ−μS/n→DN(0,1) for large n\frac{\bar{X} - \mu}{S/\sqrt{n}} \stackrel{D}{\to} N(0,1) \text{ for large } n
Xˉ±uα/2⋅Sn\bar{X} \pm u_{\alpha/2} \cdot \frac{S}{\sqrt{n}}
Rule of thumb: n≥30\text{Rule of thumb: } n \geq 30

Applications:

  • Non-normal populations
  • Unknown distribution families
  • Quick approximations
Bootstrap Confidence Intervals
Non-parametric Methods
Bootstrap methods for confidence intervals when distribution is unknown

Key Formulas:

Fn(x)=1n∑i=1nI(Xi≤x)F_n(x) = \frac{1}{n}\sum_{i=1}^n I(X_i \leq x)
θ^1∗,…,θ^B∗ from resampling\hat{\theta}^*_1, \ldots, \hat{\theta}^*_B \text{ from resampling}
[qα/2,q1−α/2] where qp=percentile of θ^∗[q_{\alpha/2}, q_{1-\alpha/2}] \text{ where } q_p = \text{percentile of } \hat{\theta}^*
[θ^−q1−α/2,θ^−qα/2][\hat{\theta} - q_{1-\alpha/2}, \hat{\theta} - q_{\alpha/2}]

Applications:

  • Complex parameter functions
  • Non-normal populations
  • Small sample scenarios
Discrete Distribution Intervals
Special Cases
Confidence intervals for discrete distribution parameters

Key Formulas:

λ∈[χ1−α/22(2T)2n,χα/22(2T+2)2n]\lambda \in \left[\frac{\chi^2_{1-\alpha/2}(2T)}{2n}, \frac{\chi^2_{\alpha/2}(2T+2)}{2n}\right]
Use Beta-F relationship for exact intervals\text{Use Beta-F relationship for exact intervals}
p^±uα/21+uα/22/np^(1−p^)n+uα/224n2\hat{p} \pm \frac{u_{\alpha/2}}{1 + u_{\alpha/2}^2/n}\sqrt{\frac{\hat{p}(1-\hat{p})}{n} + \frac{u_{\alpha/2}^2}{4n^2}}
Add ±0.5/n for continuity correction\text{Add } \pm 0.5/n \text{ for continuity correction}

Applications:

  • Count data analysis
  • Proportion estimation
  • Rate parameter inference
Standard Normal Quantiles
Critical Values
Common critical values for standard normal distribution

Key Formulas:

u0.05=1.645 (90% CI)u_{0.05} = 1.645 \text{ (90\% CI)}
u0.025=1.96 (95% CI)u_{0.025} = 1.96 \text{ (95\% CI)}
u0.005=2.576 (99% CI)u_{0.005} = 2.576 \text{ (99\% CI)}
uα/2=Φ−1(1−α/2)u_{\alpha/2} = \Phi^{-1}(1-\alpha/2)

Applications:

  • Known variance scenarios
  • Large sample approximations
  • Quick reference calculations

🔍 Quick Reference Guide

Essential decision flowchart and common critical values for confidence interval construction

Decision Flowchart
Step 1: Normal population?
→ Yes: Continue to Step 2
→ No: Use large sample (CLT) or Bootstrap
Step 2: Population variance known?
→ Yes: Use standard normal (z)
→ No: Use t-distribution
Step 3: One or two samples?
→ One: Apply single-sample formulas
→ Two: Check variance equality assumption
Common Critical Values
90% CI
z = 1.645
95% CI
z = 1.96
99% CI
z = 2.576
t(5), 95%
t = 2.571

Note: t-values depend on degrees of freedom. For large df (≥30), t ≈ z.

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