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1890 learning items

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Quiz

Independent Two-Sample Pooled t-Test

Understand the independent pooled t-test under a common population variance. Combine within-group sums of squares, calculate standard errors, test nonzero contrasts and recover signed pooled Cohen’s d. Diagnose incorrect weighting, grand-mean pooling, pairing errors and assumption-sensitive conclusions. Prerequisites: Welch comparisons, sample variances and standardized effects. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [1.3.5.3 Independent t statistics, pooled variance, effective degrees of freedom and paired design](https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm), [Standardized mean differences and within-subject designs, equations for d and difference SD](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.00863/full).

statistics

Oct 24, 2026

30 questions

Quiz

Welch's Unequal Variances Two-Sample t-Test

Use Welch’s independent two-sample t-test when population variances need not match. Calculate separate standard-error contributions, effective degrees of freedom and nonzero-null contrasts. Explore unequal sample sizes, variance pretests, allocation decisions, group transformations and the limits of unequal-variance protection. Prerequisites: one-sample t-tests and variance of independent means. The final algebraic bound is derived in the explanation. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [1.3.5.3 Independent t statistics, pooled variance, effective degrees of freedom and paired design](https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm), [Details](https://search.r-project.org/R/refmans/stats/html/t.test.html).

statistics

Oct 24, 2026

30 questions

Quiz

One-Sample Tests for Proportions & Variances

Compare inference for a single binary probability with inference for normal-population variance. Compute score Z statistics, exact binomial tails and chi-square variance pivots; distinguish standard deviation from variance benchmarks. Examine sparse expected counts, clustering, discrete tail conventions and sensitivity to nonnormality. Prerequisites: binomial and chi-square distributions and one-sample testing. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [7.2.4 Binomial model and score statistic](https://www.itl.nist.gov/div898/handbook/prc/section2/prc24.htm), [Parameters and alternative definitions](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binomtest.html), [1.3.5.8 Variance statistic and critical regions](https://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm).

statistics

Oct 24, 2026

25 questions

Quiz

One-Sample t-Test for the Population Mean

Develop one-sample t-test fluency when population standard deviation is unknown. Calculate test statistics, degrees of freedom, confidence margins and signed standardized effects, then assess Q-Q plots, outliers, selection bias and degenerate data. Later challenges connect effect size to evidence and examine unit changes and data corrections. Prerequisites: Z-tests, sample variance and confidence intervals. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [1.3.5.2 Mean limits and hypothesis test](https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm), [Standardized mean differences and within-subject designs, equations for d and difference SD](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.00863/full).

statistics

Oct 24, 2026

30 questions

Quiz

One-Sample Z-Test for the Population Mean

Practice one-sample Z-tests for a population mean with known standard deviation. Calculate signed statistics, critical sample-mean thresholds, one- and two-sided p-values, alternative distributions and sample sizes for target power. Diagnose serial dependence, transported calibration assumptions and practically small effects. Prerequisites: normal distributions, sampling variability and basic hypothesis testing. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [7.1.5 Test and confidence interval correspondence](https://www.itl.nist.gov/div898/handbook/prc/section1/prc15.htm), [7.2.2.2 One-sided sample size formula](https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm), [1.3.5.2 Mean limits and hypothesis test](https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm).

statistics

Oct 24, 2026

25 questions

Quiz

Duality Between Tests & Confidence Intervals

Connect confidence intervals with hypothesis-test decisions through matched procedures, directional bounds and inversion. Work through numerical acceptance sets, rounded boundaries, overlapping group intervals and transformations of parameters. Prerequisites: confidence intervals, test levels and p-values. Discrete tests and practical equivalence are treated as interpretation challenges, with specialist methods reserved for later topics. Includes single-select and multiselect practice with worked solutions and explanations of distractors. References: [7.1.5 Test and confidence interval correspondence](https://www.itl.nist.gov/div898/handbook/prc/section1/prc15.htm), [1.3.5.2 Mean limits and hypothesis test](https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm), [Parameters and alternative definitions](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binomtest.html).

statistics

Oct 24, 2026

15 questions

Quiz

Statistical Inference in SLR

Carry out and interpret inference for simple-regression coefficients. Calculate standard errors, t statistics, confidence intervals and ANOVA F comparisons, and reason about sampling distributions, predictor spread and coefficient covariance. Prerequisites: OLS, sums of squares and basic hypothesis testing. Applied and analytical questions stress matching assumptions, test direction, practical importance and responsible reporting. Prediction intervals are reserved for the next questionset. Further reading: [Normal regression inference](https://www.statlect.com/fundamentals-of-statistics/normal-linear-regression-model).

statistics

Oct 24, 2026

30 questions

Quiz

Partitioning Sum of Squares & Goodness-of-Fit (R-squared)

Quantify simple-regression fit through sums of squares, explained variation and residual error scales. Derive the intercept-model decomposition, connect goodness of fit to correlation, and examine constant responses, origin-constrained fits and test-sample scores. Prerequisites: OLS coefficients and residual orthogonality. Worked examples distinguish relative fit from error in meaningful units and from model validation. Further reading: [R-squared definitions](https://www.statlect.com/fundamentals-of-statistics/R-squared-of-a-linear-regression).

statistics

Oct 24, 2026

25 questions

Quiz

Ordinary Least Squares (OLS) Derivation & Properties

Derive and apply ordinary least squares for an intercept and one predictor. Calculate coefficients, audit normal equations, study centering and unit changes, and examine the scope of the BLUE theorem. Prerequisites: the SLR model, differentiation and matrix projection. Original calculations and proof checks extend to reverse regression, constrained fits and estimator-comparison limits. Further reading: [NIST least squares](https://www.itl.nist.gov/div898/handbook/pmd/section4/pmd431.htm), [Gauss–Markov theorem](https://www.statlect.com/fundamentals-of-statistics/Gauss-Markov-theorem).

statistics

Oct 24, 2026

30 questions

Quiz

The SLR Model Formulation & Assumptions

Formulate a simple linear regression model and distinguish its scientific and statistical assumptions. Interpret conditional means, errors, residuals, full rank, equal variance and error dependence, then separate Gauss–Markov conclusions from exact normal-theory inference. Prerequisites: conditional expectation, variance and the linear-algebra bridge. Scenarios emphasize exogeneity, extrapolation and careful causal language. Further reading: [Regression model assumptions](https://www.statlect.com/fundamentals-of-statistics/linear-regression), [Gauss–Markov theorem](https://www.statlect.com/fundamentals-of-statistics/Gauss-Markov-theorem).

statistics

Oct 24, 2026

20 questions

Quiz

Eigenvalues, Quadratic Forms & SVD

Explore the geometry behind covariance matrices and statistical computation. Interpret quadratic forms, eigenvectors, spectral decompositions, singular values, rank and pseudoinverses. Prerequisites: matrix arithmetic, orthogonality and least squares. Worked examples and counterexamples connect variance directions with degeneracy, minimum-norm solutions and numerical tolerance without requiring advanced PCA theory. Further reading: [Positive semidefinite matrices](https://www.statlect.com/matrix-algebra/positive-definite-matrix), [SVD reference](https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html).

statistics

Oct 24, 2026

20 questions

Quiz

Projections, Least Squares & Numerical Stability

Connect least squares to orthogonal projection, residual geometry and reliable numerical computation. Practice projector formulas, QR solving, conditioning, basis changes and the correlation bound. Prerequisites: vectors, matrix products, rank and inverses. Progress from geometric calculations to rank-deficient fits, stability tradeoffs and the limits of what numerical accuracy establishes. Further reading: [Orthogonal projection](https://textbooks.math.gatech.edu/ila/projections.html), [QR and regression](https://www.statlect.com/matrix-algebra/QR-decomposition).

statistics

Oct 24, 2026

20 questions