STAT 410 EXAM 3 VERSION C QUESTIONS & ANSWERS |
STATISTICS | UNIVERSITY OF ILLINOIS URBANA-
CHAMPAIGN | SPRING 2026 QUESTIONS AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS RATIONALES 2026
Q&A | INSTANT DOWNLOAD PDF.
Core Domains
Sufficiency, Completeness, and Minimum Variance Unbiased
Estimation
Rao-Blackwell Theorem and Lehmann-Scheffé Theorem
Neyman-Pearson Lemma and Most Powerful Tests
Uniformly Most Powerful Tests and Monotone Likelihood
Ratios
Likelihood Ratio Tests and Asymptotic Distributions
Linear Models and Least Squares Estimation
Analysis of Variance (ANOVA) and Experimental Design
Nonparametric Methods (Sign Test, Wilcoxon, Mann-Whitney)
Goodness-of-Fit Tests and Contingency Tables
Bayesian Estimation and Credible Intervals
Introduction
This comprehensive examination assesses the advanced
statistical inference knowledge required of students
completing STAT 410: Statistics and Probability II at the
University of Illinois Urbana-Champaign. The assessment
,evaluates proficiency in sufficiency, completeness, minimum
variance unbiased estimation, hypothesis testing theory,
likelihood ratio tests, linear models, nonparametric methods,
and Bayesian inference. Items are presented in multiple-
choice and scenario-based formats designed to simulate the
actual exam environment. Emphasis is placed on the
Neyman-Pearson Lemma, monotone likelihood ratio families,
Rao-Blackwellization, Lehmann-Scheffé theorem, and
asymptotic properties of test statistics. The examination
challenges candidates to integrate rigorous mathematical
derivation, statistical theory, and applied data analysis to
solve complex inference problems.
Section One: Questions 1–100
1. Let X₁, X₂, …, Xₙ be a random sample from a Bernoulli(p)
distribution. Which of the following statistics is a sufficient
statistic for p?
A. ΣXᵢ
B. X₁
C. X₁ + 2X₂
D. max(Xᵢ)
🟢 A. ΣXᵢ
,🔴 RATIONALE: By the factorization theorem, the likelihood
is L(p) = p^(ΣXᵢ)(1−p)^(n−ΣXᵢ), which factors into a function
of ΣXᵢ and p. Thus, ΣXᵢ is sufficient for p.
2. Let X₁, X₂, …, Xₙ be a random sample from a Uniform(0, θ)
distribution. Which statistic is sufficient for θ?
A. max(Xᵢ)
B. X̄
C. min(Xᵢ)
D. S²
🟢 A. max(Xᵢ)
🔴 RATIONALE: The likelihood is L(θ) = θ^(−n) for θ ≥
max(Xᵢ). By the factorization theorem, max(Xᵢ) is sufficient for
θ.
3. Which of the following best defines a complete sufficient
statistic?
A. A sufficient statistic T such that E[g(T)] = 0 for all θ implies
g(T) = 0 with probability 1
B. A sufficient statistic that is unbiased
C. A sufficient statistic with minimum variance
D. A sufficient statistic that is normally distributed
, 🟢 A. A sufficient statistic T such that E[g(T)] = 0 for all θ
implies g(T) = 0 with probability 1
🔴 RATIONALE: Completeness is a technical condition
ensuring that any unbiased estimator based on the statistic
is unique. If a complete sufficient statistic exists, it yields the
unique MVUE via the Lehmann-Scheffé theorem.
4. Let X₁, …, Xₙ be a random sample from a Poisson(λ)
distribution. Which statistic is complete and sufficient for
λ?
A. ΣXᵢ
B. X₁
C. max(Xᵢ)
D. S²
🟢 A. ΣXᵢ
🔴 RATIONALE: For a Poisson distribution, ΣXᵢ is a complete
sufficient statistic for λ. The Poisson family is a complete
sufficient statistic family.
5. The Rao-Blackwell theorem states that:
A. Conditioning an unbiased estimator on a sufficient
statistic reduces or maintains its variance
B. The MLE is always unbiased
STATISTICS | UNIVERSITY OF ILLINOIS URBANA-
CHAMPAIGN | SPRING 2026 QUESTIONS AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS RATIONALES 2026
Q&A | INSTANT DOWNLOAD PDF.
Core Domains
Sufficiency, Completeness, and Minimum Variance Unbiased
Estimation
Rao-Blackwell Theorem and Lehmann-Scheffé Theorem
Neyman-Pearson Lemma and Most Powerful Tests
Uniformly Most Powerful Tests and Monotone Likelihood
Ratios
Likelihood Ratio Tests and Asymptotic Distributions
Linear Models and Least Squares Estimation
Analysis of Variance (ANOVA) and Experimental Design
Nonparametric Methods (Sign Test, Wilcoxon, Mann-Whitney)
Goodness-of-Fit Tests and Contingency Tables
Bayesian Estimation and Credible Intervals
Introduction
This comprehensive examination assesses the advanced
statistical inference knowledge required of students
completing STAT 410: Statistics and Probability II at the
University of Illinois Urbana-Champaign. The assessment
,evaluates proficiency in sufficiency, completeness, minimum
variance unbiased estimation, hypothesis testing theory,
likelihood ratio tests, linear models, nonparametric methods,
and Bayesian inference. Items are presented in multiple-
choice and scenario-based formats designed to simulate the
actual exam environment. Emphasis is placed on the
Neyman-Pearson Lemma, monotone likelihood ratio families,
Rao-Blackwellization, Lehmann-Scheffé theorem, and
asymptotic properties of test statistics. The examination
challenges candidates to integrate rigorous mathematical
derivation, statistical theory, and applied data analysis to
solve complex inference problems.
Section One: Questions 1–100
1. Let X₁, X₂, …, Xₙ be a random sample from a Bernoulli(p)
distribution. Which of the following statistics is a sufficient
statistic for p?
A. ΣXᵢ
B. X₁
C. X₁ + 2X₂
D. max(Xᵢ)
🟢 A. ΣXᵢ
,🔴 RATIONALE: By the factorization theorem, the likelihood
is L(p) = p^(ΣXᵢ)(1−p)^(n−ΣXᵢ), which factors into a function
of ΣXᵢ and p. Thus, ΣXᵢ is sufficient for p.
2. Let X₁, X₂, …, Xₙ be a random sample from a Uniform(0, θ)
distribution. Which statistic is sufficient for θ?
A. max(Xᵢ)
B. X̄
C. min(Xᵢ)
D. S²
🟢 A. max(Xᵢ)
🔴 RATIONALE: The likelihood is L(θ) = θ^(−n) for θ ≥
max(Xᵢ). By the factorization theorem, max(Xᵢ) is sufficient for
θ.
3. Which of the following best defines a complete sufficient
statistic?
A. A sufficient statistic T such that E[g(T)] = 0 for all θ implies
g(T) = 0 with probability 1
B. A sufficient statistic that is unbiased
C. A sufficient statistic with minimum variance
D. A sufficient statistic that is normally distributed
, 🟢 A. A sufficient statistic T such that E[g(T)] = 0 for all θ
implies g(T) = 0 with probability 1
🔴 RATIONALE: Completeness is a technical condition
ensuring that any unbiased estimator based on the statistic
is unique. If a complete sufficient statistic exists, it yields the
unique MVUE via the Lehmann-Scheffé theorem.
4. Let X₁, …, Xₙ be a random sample from a Poisson(λ)
distribution. Which statistic is complete and sufficient for
λ?
A. ΣXᵢ
B. X₁
C. max(Xᵢ)
D. S²
🟢 A. ΣXᵢ
🔴 RATIONALE: For a Poisson distribution, ΣXᵢ is a complete
sufficient statistic for λ. The Poisson family is a complete
sufficient statistic family.
5. The Rao-Blackwell theorem states that:
A. Conditioning an unbiased estimator on a sufficient
statistic reduces or maintains its variance
B. The MLE is always unbiased
