Estimation Theory MCQs 2026

Larger s leads to

1. A narrower CI always ✓
2. B unchanged CI ✓
3. C zero margin of error ✓
4. D wider t-based confidence interval for μ (other things equal) ✓

An interval estimate failing to contain the true parameter in one sample

1. A proves the estimator is inconsistent ✓
2. B means α = 0 ✓
3. C implies Type II error ✓
4. D does not invalidate the 95% method — about 5% of intervals miss in the long run ✓

Relative efficiency of median to mean (normal population) is approximately

1. A 1.0 ✓
2. B 0.25 ✓
3. C 2/π ≈ 0.64 ✓
4. D π/2 ✓

A point estimate of population total is

1. A x̄ only without N ✓
2. B p̂ only ✓
3. C s²·N ✓
4. D N·x̄ under SRS if N is known ✓

Wilson score interval for π is preferred over Wald when

1. A n is small or p̂ is near 0 or 1 ✓
2. B n is extremely large only ✓
3. C σ is known ✓
4. D μ is estimated ✓

For paired data, CI for mean difference μd uses

1. A separate CIs for each group only ✓
2. B z with n1+n2−2 always ✓
3. C chi-square on 2×2 table ✓
4. D d̄ ± t_{α/2, n−1}·sd/√n ✓

Confidence level 1−α means

1. A α is coverage probability ✓
2. B α equals margin of error ✓
3. C α is sample size ✓
4. D α is the total tail probability split between two sides (two-sided) ✓

Estimate of standard error replaces unknown σ with s when

1. A always in census ✓
2. B never in surveys ✓
3. C σ is unknown and sample is from the population ✓
4. D only for proportions ✓

A consistent but biased estimator may be acceptable when

1. A n is fixed at 5 ✓
2. B bias vanishes as n grows and variance is small ✓
3. C bias grows with n ✓
4. D efficiency is zero ✓

Simultaneous confidence bands differ from single CI because they control

1. A only one parameter always ✓
2. B family-wise error over multiple parameters or points ✓
3. C only Type II error ✓
4. D only df ✓

Interval width is inversely related to

1. A n² always directly ✓
2. B confidence level decreasing ✓
3. C √n for fixed confidence and σ ✓
4. D population size always ✓

Standard error of median is generally

1. A zero ✓
2. B equal to σ ✓
3. C larger than SE of mean for normal data (asymptotically √[π/2]·σ/√n) ✓
4. D smaller than mean SE for normal ✓

Estimating σ from sample uses

1. A Σ(xi−x̄)²/n always for unbiased σ ✓
2. B s = √[Σ(xi−x̄)²/(n−1)] ✓
3. C x̄ only ✓
4. D p̂(1−p̂) ✓

If a 95% CI for μ is (10, 20), a two-sided test of H0: μ = 15 at α = 0.05 would

1. A fail to reject H0 because 15 lies inside the interval ✓
2. B reject H0 ✓
3. C be undefined ✓
4. D use chi-square only ✓

A 99% CI for μ is wider than 90% CI for same data because

1. A z_{0.005} = 2.576 > z_{0.05} = 1.645 ✓
2. B SE is larger ✓
3. C x̄ changes ✓
4. D n decreases ✓

Jackknife resampling estimates bias and variance by

1. A bootstrapping clusters only ✓
2. B using chi-square tables ✓
3. C ignoring sample size ✓
4. D leaving out one observation at a time and recomputing the estimate ✓

Asymptotic normality of MLE means

1. A MLE is biased for all n ✓
2. B MLE fails CLT ✓
3. C MLE is approximately normal with variance reaching CR bound for large n ✓
4. D MLE requires n = 2 ✓

A biased estimator can have

1. A lower mean squared error than an unbiased estimator ✓
2. B always higher MSE ✓
3. C zero variance always ✓
4. D no practical use ✓

Bayesian credible interval differs from frequentist CI because it treats

1. A the parameter as random with a posterior distribution ✓
2. B sample size as fixed forever ✓
3. C Type II error only ✓
4. D chi-square always ✓

Welch's t interval does not assume

1. A independence ✓
2. B equal population variances ✓
3. C random sampling ✓
4. D normal populations for small n ✓

Pooled variance estimate in two-sample t (equal variances) uses

1. A s1² only ✓
2. B sp² = [(n1−1)s1² + (n2−1)s2²]/(n1+n2−2) ✓
3. C (s1+s2)/2 ✓
4. D n1·s1² ✓

Confidence interval for difference μ1−μ2 (independent, equal σ, large n) is

1. A x̄1 ± z·σ ✓
2. B p̂1−p̂2 only ✓
3. C (x̄1−x̄2) ± z_{α/2}·σ_pooled·√(1/n1+1/n2) or separate SE formula ✓
4. D χ² critical only ✓

Pivotal quantity has distribution

1. A always equal to zero ✓
2. B always chi-square with unknown df ✓
3. C independent of unknown parameters ✓
4. D always the estimator itself ✓

One-sided confidence bound is used when

1. A only an upper or lower limit is needed ✓
2. B always for proportions only ✓
3. C Type I error is zero ✓
4. D chi-square test is mandatory ✓

A narrow CI with high confidence suggests

1. A a large sample and/or small population variability ✓
2. B always a biased estimator ✓
3. C always non-probability sampling ✓
4. D zero sampling error ✓

Standard error measures

1. A bias of the estimator ✓
2. B the standard deviation of the sampling distribution of an estimator ✓
3. C the confidence level ✓
4. D Type II error ✓

The sample proportion p̂ is unbiased for

1. A σ² ✓
2. B μ only ✓
3. C π ✓
4. D s² ✓

Robust estimators are

1. A always unbiased for σ² ✓
2. B always identical to x̄ ✓
3. C always maximum likelihood ✓
4. D relatively insensitive to departures from assumptions or outliers ✓

Minimum variance unbiased estimator (MVUE) is

1. A always biased ✓
2. B unbiased with smallest variance among all unbiased estimators ✓
3. C the sample median for μ always ✓
4. D the p-value ✓

The Cramér–Rao lower bound gives

1. A a lower bound on variance of unbiased estimators under regularity conditions ✓
2. B the maximum bias ✓
3. C the p-value ✓
4. D the chi-square statistic ✓

A sufficient statistic contains

1. A no information about θ ✓
2. B only non-sampling error ✓
3. C all information in the sample relevant for estimating θ ✓
4. D only the p-value ✓

Method of moments equates

1. A p-values to alpha ✓
2. B Type I and Type II errors ✓
3. C chi-square to F always ✓
4. D sample moments to theoretical moments to solve for parameters ✓

Maximum likelihood estimation (MLE) chooses θ̂ to

1. A minimize bias only ✓
2. B maximize the likelihood (or log-likelihood) of the observed data ✓
3. C maximize Type I error ✓
4. D ignore the data ✓

Sample size for proportion with planned margin E at 95% often uses conservative

1. A n = E² ✓
2. B n ≥ (z_{α/2}/E)² · 0.25 ✓
3. C n = π/E ✓
4. D n = 0.5/E ✓

Sample size for estimating μ with margin E and confidence 95% (σ known) satisfies

1. A n = σ/E ✓
2. B n = z·E/σ ✓
3. C n ≥ (z_{α/2}·σ/E)² ✓
4. D n = E²/σ ✓

A large-sample 95% CI for population proportion π is approximately

1. A p̂ ± 1.96·√[p̂(1−p̂)/n] ✓
2. B p̂ ± σ/√n ✓
3. C p̂ ± t·s ✓
4. D π ± z·σ ✓

Degrees of freedom for one-sample t interval for μ equal

1. A n ✓
2. B n − 1 ✓
3. C n + 1 ✓
4. D 2n ✓

When σ is unknown and n is small from a normal population, CI for μ uses

1. A z_{α/2} only always ✓
2. B chi-square critical values ✓
3. C t_{α/2, n−1} · s/√n ✓
4. D F critical values ✓

For a large sample with known σ, a two-sided 95% CI for μ is

1. A x̄ ± 1.96·σ/√n ✓
2. B x̄ ± 2.576·σ/√n ✓
3. C x̄ ± σ/n ✓
4. D x̄ ± 0.95·σ ✓

Increasing confidence level from 90% to 99% while n and SE fixed

1. A narrows the interval ✓
2. B leaves width unchanged ✓
3. C eliminates margin of error ✓
4. D widens the confidence interval ✓

A wider confidence interval indicates

1. A greater uncertainty about the parameter ✓
2. B a smaller standard error always ✓
3. C a larger sample size always ✓
4. D a smaller confidence level ✓

The margin of error for a large-sample CI for μ is approximately

1. A z_{α/2} · σ/√n (or t_{α/2}·s/√n when σ unknown) ✓
2. B σ·n ✓
3. C z·σ² ✓
4. D √n only ✓

A 95% confidence interval means that

1. A the parameter lies in the interval with 95% probability in one realized sample (frequentist interpretation) ✓
2. B the method used would capture the true parameter in about 95% of repeated samples ✓
3. C 95% of data fall in the interval ✓
4. D Type I error is 0.95 ✓

Consistency of an estimator means

1. A E(θ̂) = θ for every n ✓
2. B Var(θ̂) is maximum ✓
3. C θ̂ equals θ for n = 2 ✓
4. D θ̂ converges in probability to θ as sample size increases ✓

Efficiency of an estimator compares

1. A only bias ✓
2. B variances among unbiased estimators — smaller variance is more efficient ✓
3. C only consistency ✓
4. D only p-values ✓

The sample variance s² with divisor (n−1) is an unbiased estimator of

1. A μ ✓
2. B σ² ✓
3. C σ ✓
4. D π ✓

The sample mean x̄ is an unbiased estimator of

1. A the population variance σ² ✓
2. B the population median always ✓
3. C the population mean μ ✓
4. D the proportion π only ✓

An estimator is unbiased if

1. A it always equals the parameter in every sample ✓
2. B its variance is zero ✓
3. C it minimizes Type II error ✓
4. D its expected value equals the parameter: E(θ̂) = θ ✓

Interval estimation provides

1. A only a single statistic ✓
2. B only the null hypothesis ✓
3. C only the test statistic ✓
4. D a range of plausible values for a parameter with a stated confidence level ✓

Point estimation assigns

1. A a single numerical value as the estimate of a parameter ✓
2. B an interval of plausible values ✓
3. C a p-value only ✓
4. D a Type I error rate ✓