Estimation Theory MCQs 2026

Larger s leads to

1. Anarrower CI always✓
2. Bunchanged CI✓
3. Czero margin of error✓
4. Dwider t-based confidence interval for μ (other things equal)✓

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

1. Aproves the estimator is inconsistent✓
2. Bmeans α = 0✓
3. Cimplies Type II error✓
4. Ddoes 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. A1.0✓
2. B0.25✓
3. C2/π ≈ 0.64✓
4. Dπ/2✓

A point estimate of population total is

1. Ax̄ only without N✓
2. Bp̂ only✓
3. Cs²·N✓
4. DN·x̄ under SRS if N is known✓

Wilson score interval for π is preferred over Wald when

1. An is small or p̂ is near 0 or 1✓
2. Bn is extremely large only✓
3. Cσ is known✓
4. Dμ is estimated✓

For paired data, CI for mean difference μd uses

1. Aseparate CIs for each group only✓
2. Bz with n1+n2−2 always✓
3. Cchi-square on 2×2 table✓
4. Dd̄ ± t_{α/2, n−1}·sd/√n✓

Confidence level 1−α means

1. Aα is coverage probability✓
2. Bα equals margin of error✓
3. Cα is sample size✓
4. Dα is the total tail probability split between two sides (two-sided)✓

Estimate of standard error replaces unknown σ with s when

1. Aalways in census✓
2. Bnever in surveys✓
3. Cσ is unknown and sample is from the population✓
4. Donly for proportions✓

A consistent but biased estimator may be acceptable when

1. An is fixed at 5✓
2. Bbias vanishes as n grows and variance is small✓
3. Cbias grows with n✓
4. Defficiency is zero✓

Simultaneous confidence bands differ from single CI because they control

1. Aonly one parameter always✓
2. Bfamily-wise error over multiple parameters or points✓
3. Conly Type II error✓
4. Donly df✓

Interval width is inversely related to

1. An² always directly✓
2. Bconfidence level decreasing✓
3. C√n for fixed confidence and σ✓
4. Dpopulation size always✓

Standard error of median is generally

1. Azero✓
2. Bequal to σ✓
3. Clarger than SE of mean for normal data (asymptotically √[π/2]·σ/√n)✓
4. Dsmaller than mean SE for normal✓

Estimating σ from sample uses

1. AΣ(xi−x̄)²/n always for unbiased σ✓
2. Bs = √[Σ(xi−x̄)²/(n−1)]✓
3. Cx̄ only✓
4. Dp̂(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. Afail to reject H0 because 15 lies inside the interval✓
2. Breject H0✓
3. Cbe undefined✓
4. Duse chi-square only✓

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

1. Az_{0.005} = 2.576 > z_{0.05} = 1.645✓
2. BSE is larger✓
3. Cx̄ changes✓
4. Dn decreases✓

Jackknife resampling estimates bias and variance by

1. Abootstrapping clusters only✓
2. Busing chi-square tables✓
3. Cignoring sample size✓
4. Dleaving out one observation at a time and recomputing the estimate✓

Asymptotic normality of MLE means

1. AMLE is biased for all n✓
2. BMLE fails CLT✓
3. CMLE is approximately normal with variance reaching CR bound for large n✓
4. DMLE requires n = 2✓

A biased estimator can have

1. Alower mean squared error than an unbiased estimator✓
2. Balways higher MSE✓
3. Czero variance always✓
4. Dno practical use✓

Bayesian credible interval differs from frequentist CI because it treats

1. Athe parameter as random with a posterior distribution✓
2. Bsample size as fixed forever✓
3. CType II error only✓
4. Dchi-square always✓

Welch's t interval does not assume

1. Aindependence✓
2. Bequal population variances✓
3. Crandom sampling✓
4. Dnormal populations for small n✓

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

1. As1² only✓
2. Bsp² = [(n1−1)s1² + (n2−1)s2²]/(n1+n2−2)✓
3. C(s1+s2)/2✓
4. Dn1·s1²✓

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

1. Ax̄1 ± z·σ✓
2. Bp̂1−p̂2 only✓
3. C(x̄1−x̄2) ± z_{α/2}·σ_pooled·√(1/n1+1/n2) or separate SE formula✓
4. Dχ² critical only✓

Pivotal quantity has distribution

1. Aalways equal to zero✓
2. Balways chi-square with unknown df✓
3. Cindependent of unknown parameters✓
4. Dalways the estimator itself✓

One-sided confidence bound is used when

1. Aonly an upper or lower limit is needed✓
2. Balways for proportions only✓
3. CType I error is zero✓
4. Dchi-square test is mandatory✓

A narrow CI with high confidence suggests

1. Aa large sample and/or small population variability✓
2. Balways a biased estimator✓
3. Calways non-probability sampling✓
4. Dzero sampling error✓

Standard error measures

1. Abias of the estimator✓
2. Bthe standard deviation of the sampling distribution of an estimator✓
3. Cthe confidence level✓
4. DType II error✓

The sample proportion p̂ is unbiased for

1. Aσ²✓
2. Bμ only✓
3. Cπ✓
4. Ds²✓

Robust estimators are

1. Aalways unbiased for σ²✓
2. Balways identical to x̄✓
3. Calways maximum likelihood✓
4. Drelatively insensitive to departures from assumptions or outliers✓

Minimum variance unbiased estimator (MVUE) is

1. Aalways biased✓
2. Bunbiased with smallest variance among all unbiased estimators✓
3. Cthe sample median for μ always✓
4. Dthe p-value✓

The Cramér–Rao lower bound gives

1. Aa lower bound on variance of unbiased estimators under regularity conditions✓
2. Bthe maximum bias✓
3. Cthe p-value✓
4. Dthe chi-square statistic✓

A sufficient statistic contains

1. Ano information about θ✓
2. Bonly non-sampling error✓
3. Call information in the sample relevant for estimating θ✓
4. Donly the p-value✓

Method of moments equates

1. Ap-values to alpha✓
2. BType I and Type II errors✓
3. Cchi-square to F always✓
4. Dsample moments to theoretical moments to solve for parameters✓

Maximum likelihood estimation (MLE) chooses θ̂ to

1. Aminimize bias only✓
2. Bmaximize the likelihood (or log-likelihood) of the observed data✓
3. Cmaximize Type I error✓
4. Dignore the data✓

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

1. An = E²✓
2. Bn ≥ (z_{α/2}/E)² · 0.25✓
3. Cn = π/E✓
4. Dn = 0.5/E✓

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

1. An = σ/E✓
2. Bn = z·E/σ✓
3. Cn ≥ (z_{α/2}·σ/E)²✓
4. Dn = E²/σ✓

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

1. Ap̂ ± 1.96·√[p̂(1−p̂)/n]✓
2. Bp̂ ± σ/√n✓
3. Cp̂ ± t·s✓
4. Dπ ± z·σ✓

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

1. An✓
2. Bn − 1✓
3. Cn + 1✓
4. D2n✓

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

1. Az_{α/2} only always✓
2. Bchi-square critical values✓
3. Ct_{α/2, n−1} · s/√n✓
4. DF critical values✓

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

1. Ax̄ ± 1.96·σ/√n✓
2. Bx̄ ± 2.576·σ/√n✓
3. Cx̄ ± σ/n✓
4. Dx̄ ± 0.95·σ✓

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

1. Anarrows the interval✓
2. Bleaves width unchanged✓
3. Celiminates margin of error✓
4. Dwidens the confidence interval✓

A wider confidence interval indicates

1. Agreater uncertainty about the parameter✓
2. Ba smaller standard error always✓
3. Ca larger sample size always✓
4. Da smaller confidence level✓

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

1. Az_{α/2} · σ/√n (or t_{α/2}·s/√n when σ unknown)✓
2. Bσ·n✓
3. Cz·σ²✓
4. D√n only✓

A 95% confidence interval means that

1. Athe parameter lies in the interval with 95% probability in one realized sample (frequentist interpretation)✓
2. Bthe method used would capture the true parameter in about 95% of repeated samples✓
3. C95% of data fall in the interval✓
4. DType I error is 0.95✓

Consistency of an estimator means

1. AE(θ̂) = θ for every n✓
2. BVar(θ̂) is maximum✓
3. Cθ̂ equals θ for n = 2✓
4. Dθ̂ converges in probability to θ as sample size increases✓

Efficiency of an estimator compares

1. Aonly bias✓
2. Bvariances among unbiased estimators — smaller variance is more efficient✓
3. Conly consistency✓
4. Donly p-values✓

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

1. Aμ✓
2. Bσ²✓
3. Cσ✓
4. Dπ✓

The sample mean x̄ is an unbiased estimator of

1. Athe population variance σ²✓
2. Bthe population median always✓
3. Cthe population mean μ✓
4. Dthe proportion π only✓

An estimator is unbiased if

1. Ait always equals the parameter in every sample✓
2. Bits variance is zero✓
3. Cit minimizes Type II error✓
4. Dits expected value equals the parameter: E(θ̂) = θ✓

Interval estimation provides

1. Aonly a single statistic✓
2. Bonly the null hypothesis✓
3. Conly the test statistic✓
4. Da range of plausible values for a parameter with a stated confidence level✓

Point estimation assigns

1. Aa single numerical value as the estimate of a parameter✓
2. Ban interval of plausible values✓
3. Ca p-value only✓
4. Da Type I error rate✓