Bivariate Correlated Pair Generator
Generates dependent pairs (X_1, X_2) with controlled inter-subject correlation ρ = 0.80 using Cholesky factorization.
Statistics & Probability / Paired-Samples t-Test
FREE TO EXPLORELABORATORY 28 · STATISTICS & PROBABILITY
Paired-Samples t-Test, Dependent Observations, and Within-Subject Difference Analysis
Investigate paired-sample experimental designs, analyze dependent bivariate observations (before/after trials, matched case-controls), eliminate between-subject variance via within-subject difference scores, and quantify test power enhancement. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED PAIRED T-STATISTIC E[T_D]—
SAMPLING TELEMETRY
Ready to sample with Factor k = 1.0.
Adjust Factor k, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
The paired design loses half its potential degrees of freedom (n - 1 versus 2n - 2) relative to independent groups. The empirical regression of expected paired t-statistics against treatment factor k confirms theoretical variance compression with slope m = 5.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Within-Subject Difference Transformer
Computes paired difference scores D_i = X_{2, i} - X_{1, i} to eliminate inter-subject baseline variability.
Independent Two-Sample Comparator
Computes independent two-sample t on the same data to illustrate the dramatic loss of power when pairing is ignored.
Paired Standard Error Resolver
Calculates variance of differences Var(D) = σ_1^2 + σ_2^2 - 2ρ σ_1 σ_2 and evaluates standard error s_D / √n.
Paired t-Statistic Engine
Evaluates paired t-statistic t_D = (D̄ - μ_D) / (s_D / √n) with n - 1 degrees of freedom.
Paired Precision Regression Unit
Performs linear regression of expected paired t versus treatment factor k to confirm the slope m = 5.000.
FOLLOW THE METHOD
A reason behind every step.
Investigate paired-sample experimental designs, analyze dependent bivariate observations (before/after trials, matched case-controls), eliminate between-subject variance via within-subject difference scores, and quantify test power enhancement. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Configure Bivariate Correlation and Sample Size
Set sample size to n = 25 matched pairs and establish positive inter-pair correlation ρ = 0.80 on the paired design console.
02
Synthesize Baseline and Post-Treatment Observations
Generate paired observations (X_{1, i}, X_{2, i}) where X_{2, i} = X_{1, i} + Δ + ε_i, modeling high between-subject baseline variance.
03
Compute Within-Subject Difference Scores
Transform pairs into single difference variables D_i = X_{2, i} - X_{1, i}. Observe that subtracting removes subject baselines entirely.
04
Compare Variance Reduction against Independent Design
Verify that Var(D) = σ_1^2 + σ_2^2 - 2ρ σ_1 σ_2 = 1 + 1 - 2(0.80) = 0.40, achieving a 5-fold variance compression relative to independent designs.
05
Step Treatment Effect Across Factor k = 1 to 6
Increment the treatment effect factor across standardized settings k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording expected paired t-statistics.
06
Conduct Linear Regression on Paired Test Power
Fit a linear regression model y = mx to coordinate pairs (x = k, y = E[t_D]). Confirm that empirical slope m = 5.0000 matches theoretical paired precision.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Paired-Samples t-Test mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 5.0000 · x
| # | Treatment Effect Factor k (factor) | Expected Paired t-Statistic E[t_D] (Statistic E[t_D]) | Treatment Effect Factor k | Expected Paired t-Statistic E[t_D] | Remove |
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