Interval Estimation

Flashcards for key vocabulary and concepts related to Interval Estimation.

Interval Estimation

A statistical technique used to estimate a population parameter by specifying a range, or interval, of plausible values.

Interval Estimator

The random interval [L(X), U(X)], where L(X) and U(X) are functions of a sample that satisfy L(x) <= U(x) for all x in the sample space.

Coverage Probability

The probability that the random interval [L(X), U(X)] covers the true parameter θ, denoted by Pθ(θ ∈ [L(X), U(X)]) or P(θ ∈ [L(X), U(X)] | θ).

Confidence Coefficient

The infimum of the coverage probabilities, inf Pθ(θ ∈ [L(X), U(X)]).

Confidence Interval

Interval estimators, together with a measure of confidence (usually a confidence coefficient).

General Format of a Confidence Interval

Point Estimate ± Margin of Error, where Margin of Error E = (tabulated value) x (standard error).

Confidence Interval for Population Mean (σ known)

x̄ ± Zα/2 * (σ/√n)

Confidence Interval for Population Mean (σ unknown, n ≥ 30)

x̄ ± Zα/2 * (s/√n)

Confidence Interval for Population Mean (σ unknown, n < 30)

x̄ ± tα/2, n-1 * (s/√n)

Confidence Interval for Difference between Two Population Means (σ1^2 and σ2^2 known)

(x̄1 - x̄2) ± Zα/2 * √(σ1^2/n1 + σ2^2/n2)

Confidence Interval for Difference between Two Population Means (σ1^2 and σ2^2 unknown, n1 ≥ 30 and n2 ≥ 30)

(x̄1 - x̄2) ± Zα/2 * √(s1^2/n1 + s2^2/n2)

Confidence Interval for Difference between Two Population Means (σ1^2 = σ2^2 unknown, n1 < 30 and n2 < 30)

(x̄1 - x̄2) ± t(α/2, v) * Sp * √(1/n1 + 1/n2), where Sp = √(((n1-1)s1^2 + (n2-1)s2^2)/(n1+n2-2)) and v = n1 + n2 - 2

Confidence Interval for Paired Observations (μ1 - μ2 = μd)

x̄d ± tα/2, v * (Sd/√n), where v = n-1

Inverting a Test Statistic

A method used in interval estimation where a confidence interval is constructed by reversing the logic of a hypothesis test.

Pivotal Quantity (or Pivot)

A random variable Q(X, θ) = Q(X1, …, Xn, θ) is a pivotal quantity if the distribution of Q(X, θ) is independent of all parameters.

Normal Pivotal Quantity (σ^2 known)

Q = (X̄ - μ) / (σ/√n) ~ N(0, 1)

Normal Pivotal Quantity (σ^2 unknown)

Q = (X̄ - μ) / (S/√n) ~ t(n-1)

Normal Pivotal Quantity (μ is known)

Q = Σ(Xi - μ)^2 / σ^2 ~ χ^2(n)

Normal Pivotal Quantity (μ is unknown and σ^2 is known)

Q = (n-1)S^2 / σ^2 ~ χ^2(n-1)

Effect of Sample Size on Interval Length

As n increases, interval length decreases, and the estimate becomes more precise.

Effect of Confidence Level on Interval Length

Higher Confidence: safer, more cautious estimates → less precision. Lower Confidence: narrower, more precise intervals → more risk of missing the true value.