Interval Estimation Flashcards

Flashcards based on lecture notes about Interval Estimation

Interval Estimation

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

Confidence Level

Quantifies the degree of certainty in the estimate.

Interval Estimator

The random interval [L(X), U(X)].

Coverage Probability

The probability that the random interval [L(X), U(X)] covers the true parameter θ.

Confidence Coefficient

The infimum of the coverage probabilities inf P(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 E = (tabulated value) * (standard error).

Confidence Interval for Population Mean (σ known)

x̄ ± Z(α/2) * (σ/√n)

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

x̄ ± Z(α/2) * (s/√n)

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

x̄ ± t(α/2, n-1) * (s/√n)

Confidence Interval for Difference Between Two Population Means (σ1² and σ2² known)

(x̄₁ - x̄₂) ± Z(α/2) * √(σ₁²/n₁ + σ₂²/n₂)

Confidence Interval for Difference Between Two Population Means (σ1² and σ2² unknown, n1≥30 and n2≥30)

(x̄₁ - x̄₂) ± Z(α/2) * √(s₁²/n₁ + s₂²/n₂)

Confidence Interval for Difference Between Two Population Means (σ1² = σ2² are unknown, and n1 < 30 and n2 < 30)

(x̄₁ - x̄₂) ± t(α/2, v) * Sp * √(1/n₁ + 1/n₂), where Sp is the pooled standard deviation and v = n1 + n2 - 2.

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

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

Inverting a Test Statistic

A method used in interval estimation to construct a confidence interval by reversing the logic of a hypothesis test.

Pivotal Quantity

A random variable Q(X, θ) whose distribution is independent of all parameters.

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 -> wider, more cautious estimates -> less precision. Lower Confidence -> narrower, more precise intervals -> more risk of missing true value.