MathIsimple
Back to Mathematical Statistics
Interval Estimation
6-8 Hours

Confidence Intervals & Interval Estimation

Master the theory and practice of interval estimation: from pivotal quantities to optimal interval construction

Fundamental Concepts

Core definitions and principles of interval estimation

Interval Estimator

An interval estimator for parameter θ is a pair of statistics [θ̂_L(X̃), θ̂_U(X̃)] such that θ̂_L ≤ θ̂_U.

[θ^L(X~),θ^U(X~)][\hat{\theta}_L(\tilde{X}), \hat{\theta}_U(\tilde{X})]
Coverage Probability

The probability that the random interval contains the fixed parameter θ.

C(θ)=Pθ{θ∈[θ^L,θ^U]}C(\theta) = P_{\theta}\{\theta \in [\hat{\theta}_L, \hat{\theta}_U]\}
Confidence Coefficient

The minimum coverage probability over all possible values of θ.

γ=inf⁡θ∈ΘC(θ)\gamma = \inf_{\theta \in \Theta} C(\theta)
Correct Interpretation (Critical!)
  • Correct: "If we repeat sampling many times, ~95% of the constructed intervals will contain θ"
  • Incorrect: "There's a 95% probability that θ lies in this specific interval"

The parameter θ is fixed; the interval is random. After observing data, θ is either in the interval or not.

Pivotal Quantity Method

The fundamental technique for constructing exact confidence intervals

Definition
A pivotal quantity G(X̃, θ) is a function of data and parameter whose distribution is completely known and independent of any unknown parameters.
G(X~,θ)∼FGwhere FG is known and does not depend on θG(\tilde{X}, \theta) \sim F_G \quad \text{where } F_G \text{ is known and does not depend on } \theta

Normal Mean (σ² known)

Xˉ−μσ/n∼N(0,1)\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)

Normal Mean (σ² unknown)

Xˉ−μS/n∼t(n−1)\frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n-1)

Normal Variance

(n−1)S2σ2∼χ2(n−1)\frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)

Variance Ratio

S12/σ12S22/σ22∼F(n1−1,n2−1)\frac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2} \sim F(n_1-1, n_2-1)
Construction Steps
Systematic procedure for building CIs using pivots
  1. 1

    Find a Pivotal Quantity

    Identify G(X̃, θ) whose distribution is known and independent of unknown parameters.

  2. 2

    Determine Critical Values

    Find quantiles c and d such that P(c ≤ G ≤ d) = 1-α.

  3. 3

    Algebraic Inversion

    Solve the inequality c ≤ G(X̃, θ) ≤ d for θ.

  4. 4

    Extract Interval Bounds

    The resulting bounds [θ̂_L(X̃), θ̂_U(X̃)] form the CI.

Normal Population Confidence Intervals

Exact CIs for mean and variance of normal populations

Mean CI (Variance Known)

For X₁, ..., Xₙ ~ N(μ, σ²) with σ² known:

[Xˉ−zα/2⋅σn,Xˉ+zα/2⋅σn]\left[\bar{X} - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}, \quad \bar{X} + z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\right]

Pivotal quantity: Z = (X̄ - μ)/(σ/√n) ~ N(0,1)

Mean CI (Variance Unknown)

For X₁, ..., Xₙ ~ N(μ, σ²) with σ² unknown:

[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}}, \quad \bar{X} + t_{\alpha/2}(n-1) \cdot \frac{S}{\sqrt{n}}\right]

Pivotal quantity: T = (X̄ - μ)/(S/√n) ~ t(n-1)

Variance CI

For X₁, ..., Xₙ ~ N(μ, σ²):

[(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)}, \quad \frac{(n-1)S^2}{\chi^2_{1-\alpha/2}(n-1)}\right]

Pivotal quantity: (n-1)S²/σ² ~ χ²(n-1)

Examples and Solutions

Practical applications of confidence interval construction

Example 1: Normal Mean CI (Known Variance)

Problem: Given X₁, ..., Xₙ ~ N(μ, 16), n = 25, x̄ = 50, construct a 95% CI for μ.

Solution:

Since σ² = 16 is known, σ = 4. The 95% CI is:

x̄ ± z₀.₀₂₅ × σ/√n = 50 ± 1.96 × 4/√25 = 50 ± 1.96 × 0.8 = 50 ± 1.568

Final CI: [48.43, 51.57]

Example 2: Normal Mean CI (Unknown Variance)

Problem: Given X₁, ..., Xₙ ~ N(μ, σ²) with σ² unknown, n = 20, x̄ = 45, s = 5, construct a 90% CI for μ.

Solution:

Since σ² is unknown, use t-distribution. For 90% CI with n-1 = 19 df, t₀.₀₅(19) ≈ 1.729.

x̄ ± t₀.₀₅(19) × s/√n = 45 ± 1.729 × 5/√20 = 45 ± 1.729 × 1.118 = 45 ± 1.933

Final CI: [43.07, 46.93]

Example 3: Variance CI

Problem: Given X₁, ..., Xₙ ~ N(μ, σ²), n = 15, s² = 25, construct a 95% CI for σ².

Solution:

Use chi-square distribution. For 95% CI with n-1 = 14 df: χ²₀.₀₂₅(14) ≈ 26.12, χ²₀.₉₇₅(14) ≈ 5.63.

Lower bound: (n-1)s²/χ²₀.₀₂₅ = 14×25/26.12 ≈ 13.40

Upper bound: (n-1)s²/χ²₀.₉₇₅ = 14×25/5.63 ≈ 62.17. Final CI: [13.40, 62.17]

Practice Quiz

Test your understanding with 10 multiple-choice questions

Practice Quiz
10
Questions
0
Correct
0%
Accuracy
1
What does a 95% confidence level mean for a confidence interval [θ̂_L, θ̂_U]?
2
For a normal population N(μ, σ²) with unknown variance, which pivotal quantity should be used to construct a confidence interval for μ?
3
Given: X ~ N(μ, 16), n = 25, x̄ = 50, 95% confidence level (z₀.₀₂₅ = 1.96). What is the 95% confidence interval for μ?
4
All else being equal, what happens to the width of a confidence interval when the sample size increases from n = 16 to n = 64?
5
Under what condition should you use a t-distribution instead of a standard normal distribution for constructing confidence intervals?
6
For comparing means of two normal populations with equal but unknown variances, the appropriate confidence interval for μ₁ - μ₂ uses which distribution?
7
What is the relationship between confidence coefficient and confidence level?
8
If θ̂_L is a one-sided confidence lower bound with confidence level 1-α, what does this mean?
9
Bootstrap confidence intervals are particularly useful when:
10
A researcher reports a 95% confidence interval for mean income as [45,000,45,000, 55,000]. Which interpretation is correct?

Frequently Asked Questions

Common questions about confidence intervals

How should I correctly interpret a 95% confidence interval?
Correct: If we repeat the sampling procedure many times, about 95% of the constructed intervals will contain the true parameter θ. Incorrect: 'There's a 95% probability that θ is in this interval' — once computed, θ is either in the interval or not.
Key Point: The parameter is fixed; the interval is random
When should I use t-distribution vs z-distribution?
Use z-distribution when σ² is known. Use t-distribution when σ² is unknown and estimated from data. The t-distribution has heavier tails, giving wider intervals to account for the extra uncertainty. For large n ≥ 30, they are nearly identical.
Use t when σ2 is unknown\text{Use } t \text{ when } \sigma^2 \text{ is unknown}
What is a pivotal quantity and why is it useful?
A pivotal quantity G(X̃, θ) has a distribution that doesn't depend on unknown parameters. This allows us to make exact probability statements like P{c ≤ G ≤ d} = 1-α and then invert to get confidence intervals. Examples: (X̄-μ)/(S/√n) ~ t(n-1).
G(X~,θ)∼FG (known, independent of θ)G(\tilde{X}, \theta) \sim F_G \text{ (known, independent of } \theta \text{)}
What's the relationship between CI and hypothesis testing?
They are duals: A 1-α CI contains exactly those θ₀ values that would not be rejected in a level-α test of H₀: θ = θ₀. If θ₀ ∈ CI, fail to reject; if θ₀ ∉ CI, reject. CIs provide more information than just reject/fail to reject.
Key Point: CI and hypothesis test are mathematically equivalent
How does sample size affect the confidence interval?
Larger sample size n leads to narrower intervals without reducing confidence level. The width is proportional to 1/√n, so quadrupling n halves the width. This is the only way to get both precision and reliability simultaneously.
Width∝1n\text{Width} \propto \frac{1}{\sqrt{n}}
Ask AI ✨
MathIsimple – Simple, Friendly Math Tools & Learning