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1890 learning items

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Quiz

Unbiasedness, Bias & Mean Squared Error (MSE)

Build a rigorous understanding of unbiasedness, bias & mean squared error (mse) through 25 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: the earlier core Statistics curriculum, including probability distributions, expectation, calculus and the preceding mathematical-statistics sets. Competencies: Distinguish estimators, estimates and bias; Prove unbiasedness of means and sample variance; Derive the MSE bias-variance decomposition; Compute risk and compare biased and unbiased estimators; Distinguish known and estimated mean variance formulas; Study shrinkage and parameter-dependent risk; Relate MSE consistency to probability convergence; Identify fourth-moment requirements for variance-estimation risk. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [Scalar bias-variance decomposition](https://www.statlect.com/glossary/mean-squared-error), [Unbiased and biased sample variance](https://www.statlect.com/glossary/sample-variance), [Point estimation, bias and risk](https://hanbaeklyu.com/wp-content/uploads/2020/10/lecturenote_170s.pdf).

statistics

Oct 26, 2026

25 questions

Quiz

Continuous Mapping, Slutsky & Delta Method

Build a rigorous understanding of continuous mapping, slutsky & delta method through 20 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: the earlier core Statistics curriculum, including probability distributions, expectation, calculus and the preceding mathematical-statistics sets. Competencies: Apply continuous mapping with continuity checks; Apply Slutsky with a constant probability limit; Derive scalar delta-method variance; Propagate multivariate covariance using a Jacobian; Studentize with consistent scale estimation; Identify zero-derivative and boundary failures; Use second-order scaling when needed; Diagnose discontinuities and plug-in standard-error errors. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [Proof and multivariate generalization](https://www.statlect.com/asymptotic-theory/delta-method), [Constant probability limits and joint transformations](https://www.statlect.com/asymptotic-theory/Slutsky-theorem), [Asymptotic normality and delta method](https://hanbaeklyu.com/wp-content/uploads/2020/10/lecturenote_170s.pdf).

statistics

Oct 26, 2026

20 questions

Quiz

Convergence, Inequalities & Limit Theorem Proofs

Build a rigorous understanding of convergence, inequalities & limit theorem proofs through 20 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: the earlier core Statistics curriculum, including probability distributions, expectation, calculus and the preceding mathematical-statistics sets. Competencies: Define modes of stochastic convergence; Apply Markov, Chebyshev and Jensen inequalities; Prove a finite-variance weak law and derive bounds; Construct rare-event counterexamples for moment convergence; Distinguish marginal weak convergence and pathwise closeness; Develop a characteristic-function CLT proof sketch; State finite-moment and MGF assumptions precisely; Use almost sure convergence criteria and distinguish proof sufficiency. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [Convergence, inequalities, laws of large numbers and CLT](https://www.statslab.cam.ac.uk/~mike/probability/chapter1-2.pdf), [Lindeberg–Levy iid CLT and finite variance](https://www.statlect.com/asymptotic-theory/central-limit-theorem), [Definition and implications](https://www.statlect.com/asymptotic-theory/convergence-in-probability).

statistics

Oct 26, 2026

20 questions

Quiz

Deriving Distributions via MGFs & Characteristic Functions

Build a rigorous understanding of deriving distributions via mgfs & characteristic functions through 20 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: the earlier core Statistics curriculum, including probability distributions, expectation, calculus and the preceding mathematical-statistics sets. Competencies: Introduce complex numbers and Euler decomposition; Prove characteristic-function existence and bounds; Derive discrete-distribution transforms from sums; Derive continuous-distribution transforms from integrals; Identify independent sum distributions using uniqueness; Transform characteristic functions under scaling and shifts; Distinguish moment existence from transform existence; Diagnose invalid dependence and support arguments. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [Definition, existence, uniqueness and transform properties](https://www.statlect.com/fundamentals-of-probability/characteristic-function), [Exponential integral and independent sums](https://www.statlect.com/fundamentals-of-probability/moment-generating-function), [Distribution families and transforms](https://hanbaeklyu.com/wp-content/uploads/2020/10/lecturenote_170s.pdf).

statistics

Oct 26, 2026

20 questions

Quiz

Moment-Generating Functions (MGFs)

Build a rigorous understanding of moment-generating functions (mgfs) through 30 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: the earlier core Statistics curriculum, including distributions, expectation, calculus and independent random variables. Competencies: Define MGFs and their existence domains; Extract raw moments and variance from derivatives; Use affine transformations and argument scaling; Combine independent transforms and diagnose dependence; Use uniqueness without overclaiming moment identification; Distinguish bounded laws and heavy-tail existence failures; Derive log-transform cumulants and sample-mean scaling; Evaluate transform identities, symmetry and convergence restrictions. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [Definition, moments, uniqueness, linear transformations and independent sums](https://www.statlect.com/fundamentals-of-probability/moment-generating-function), [Theorems 1.28 and 1.30; MGF existence and uniqueness](https://www.statslab.cam.ac.uk/~mike/probability/chapter1-2.pdf).

statistics

Oct 26, 2026

30 questions

Quiz

Nonresponse, Calibration & Survey Uncertainty

Build a rigorous understanding of nonresponse, calibration & survey uncertainty through 20 original single-select and multiselect questions. Begin with core definitions, then work through calculations, assumptions, counterexamples and advanced diagnostic challenges. Prerequisites: probability sampling, weighted means and elementary variance estimation. Competencies: Distinguish frame coverage, unit nonresponse and item nonresponse; Assess response-rate and auxiliary-variable evidence; Compute poststratification and inverse-response weights; Explain marginal calibration and its support limits; Diagnose weight variability and design effects; Compute and interpret replicate-weight uncertainty; Evaluate trimming and adjustment uncertainty; Assess assumptions for nonprobability generalization. Every item includes a worked explanation and reasons for rejecting distractors. Further reading: [D3: coverage and nonresponse indicators](https://www.census.gov/about/policies/quality/standards/standardd3.html), [Calibration and nonresponse adjustment](https://www.census.gov/library/working-papers/2010/adrm/rrs2010-03.html), [Replicate weights and repeated weighting adjustments](https://www2.census.gov/programs-surveys/nychvs/technical-documentation/variances/2017-NYCHVS-Guide-to-Estimating-Variances.pdf), [Sampling frames, probability and nonprobability recruitment](https://aapor.org/standards-and-ethics/best-practices/).

statistics

Oct 26, 2026

20 questions

Quiz

Kolmogorov-Smirnov (K-S) Tests

Compare empirical and reference distributions using one- and two-sample Kolmogorov–Smirnov methods. Calculate maximum CDF gaps and examine parameter estimation, Lilliefors calibration, ties, transformations, dependence and practical interpretation. Prerequisites: CDFs, ordered samples and categorical goodness-of-fit reasoning. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

20 questions

Quiz

Chi-Square Test of Homogeneity & Small-Sample Exact Tests

Match categorical inference to separate samples and paired measurements. Compare homogeneity and independence designs, calculate hypergeometric probabilities, inspect Yates’ correction and use asymptotic or exact McNemar inference for binary transitions. Prerequisites: Independence tables, conditional probability and paired data. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

25 questions

Quiz

Chi-Square Test of Independence

Analyze two-way count tables using the chi-square test of independence. Move from fitted counts and degrees of freedom to adjusted residuals, omnibus interpretation and design problems involving sparse support, clustering, weighting and missing data. Prerequisites: Categorical tables and chi-square goodness of fit. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

30 questions

Quiz

Chi-Square Goodness-of-Fit Test

Assess specified categorical distributions with Pearson goodness-of-fit methods. Calculate expected counts, contributions and degrees of freedom; investigate sparse categories, parameter fitting, pooling, simulation calibration and limitations of nonrejection. Prerequisites: Probability, categorical counts and hypothesis tests. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

25 questions

Quiz

Equivalence, Noninferiority & Prospective Power

Plan studies around meaningful effects and acceptable differences. Practice equivalence and noninferiority hypotheses, TOST and matching confidence intervals, prospective sample-size calculations, simulation-based power and sensitivity to planning assumptions. Prerequisites: One-sided tests, confidence intervals, variance and error rates. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

20 questions

Quiz

Multiple Hypothesis Testing & False Discovery Rate (FDR)

Build responsible multiple-testing decisions for families of claims and discovery screens. Work through Bonferroni, Holm and Benjamini–Hochberg calculations, distinguish FWER from FDR, and evaluate dependence assumptions, q-values and selective reporting. Prerequisites: Individual p-values, probability and research-integrity concepts. Includes single-select and multiselect questions with worked solutions and distractor explanations. The saved learning sequence moves from introductory understanding through application to analytical and synthesis challenges; each question is standalone.

statistics

Oct 26, 2026

25 questions