Heteroscedastic Dual-Sample Generator
Generates independent two-group Gaussian datasets with disparate sample sizes (n_1, n_2) and unequal variances (σ_1^2, σ_2^2).
Statistics & Probability / Two-Sample Welch t-Test
FREE TO EXPLORELABORATORY 27 · STATISTICS & PROBABILITY
Two-Sample Welch's t-Test for Unequal Variances and Satterthwaite Degrees of Freedom
Investigate two-sample hypothesis testing under variance heteroscedasticity, formulate Welch's t-statistic, evaluate Satterthwaite's effective degrees of freedom approximation, and verify two-sample mean difference detection. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED WELCH TEST STATISTIC E[T]—
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
Welch's t-test assumes normal parent distributions; severe departures from normality with small samples require rank tests. The empirical regression of expected Welch test statistics against difference factor k confirms robust two-sample scaling with slope m = 2.5000 and R² = 1.000000.
Statistical tools & sampling modules.
Pooled Two-Sample Student's t Engine
Computes classical Student's two-sample t assuming equal variances to demonstrate Type I error inflation.
Welch-Satterthwaite t-Statistic Resolver
Computes Welch's unpooled test statistic t = (x̄_1 - x̄_2) / √(s_1^2/n_1 + s_2^2/n_2).
Satterthwaite Degrees of Freedom Calculator
Evaluates the fractional effective degrees of freedom ν_W via the Welch-Satterthwaite equation.
Behrens-Fisher Problem Simulator
Simulates 20,000 heteroscedastic trials to compare Welch versus Student empirical rejection rates under the null.
Mean Difference Regression Unit
Performs linear regression of expected Welch t versus difference factor k to confirm the gradient m = 2.500.
FOLLOW THE METHOD
A reason behind every step.
Investigate two-sample hypothesis testing under variance heteroscedasticity, formulate Welch's t-statistic, evaluate Satterthwaite's effective degrees of freedom approximation, and verify two-sample mean difference detection. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Set Heteroscedastic Variance and Sample Size Parameters
Configure group 1 with n_1 = 10, σ_1^2 = 1.0 and group 2 with n_2 = 25, σ_2^2 = 4.0 on the two-sample simulation interface.
02
Synthesize Two Independent Gaussian Groups under H_0
Draw independent samples with equal means μ_1 = μ_2 = 0.0. Compute sample means (x̄_1, x̄_2) and sample variances (s_1^2, s_2^2).
03
Calculate Satterthwaite Effective Degrees of Freedom
Evaluate the Welch-Satterthwaite equation to obtain effective degrees of freedom ν_W, noting that ν_W is non-integer and bounded between min(n_1-1, n_2-1) and n_1+n_2-2.
04
Evaluate Classical Pooled t versus Welch t
Compare pooled Student's t with unpooled Welch's t across 20,000 runs, observing that Student's t inflates Type I error when smaller samples have larger variances.
05
Step Mean Difference Across Factor k = 1 to 6
Increment the true mean difference Δμ = μ_1 - μ_2 across standardized settings k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording expected Welch t-statistics.
06
Perform Linear Regression on Two-Sample Response
Fit a linear regression model y = mx to coordinate pairs (x = k, y = E[t]). Confirm that empirical slope m = 2.5000 validates Welch test scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Two-Sample Welch t-Test mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.5000 · x
| # | True Difference Factor k (factor) | Expected Welch Test Statistic E[t] (Welch t) | True Difference Factor k | Expected Welch Test Statistic E[t] | Remove |
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