Null and Alternative Data Generator
Generates Gaussian sample datasets under both null H_0: μ = μ_0 and alternative H_1: μ = μ_0 + δ σ hypotheses.
Statistics & Probability / Single-Sample Z & t Tests
FREE TO EXPLORELABORATORY 26 · STATISTICS & PROBABILITY
One-Sample Hypothesis Testing: Z-Tests, Student's t-Tests, Critical Values, and p-Values
Formulate null and alternative statistical hypotheses, evaluate Neyman-Pearson decision rules, compute one-sample Z-statistics and Student's t-statistics, calculate exact p-values, and verify the linear scaling of the non-centrality parameter. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
EXPECTED TEST STATISTIC E[Z]—
SAMPLING TELEMETRY
Ready to sample with Effect Size δ = 1.0.
Adjust Effect Size δ, 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
Statistical significance does not imply practical importance; very large sample sizes can yield tiny p-values for trivial effects. The empirical regression of expected test statistics against effect size confirms the theoretical non-centrality shift with slope m = 2.0000 and R² = 1.000000.
Statistical tools & sampling modules.
One-Sample Z-Test Calculator
Computes exact Z-statistics Z = (x̄ - μ_0) / (σ / √n) when population variance is known.
One-Sample Student's t-Test Unit
Computes Student's t-statistic T = (x̄ - μ_0) / (s / √n) with unknown variance and n - 1 degrees of freedom.
Exact p-Value Engine
Calculates one-tailed and two-tailed p-values via standard normal and Student's t cumulative distribution functions.
Type I and Type II Error Simulator
Simulates 20,000 hypothesis test replications to verify nominal significance level α and empirical test power 1 - β.
Non-Centrality Linear Regression Unit
Performs linear regression of expected test statistic versus effect size δ to confirm slope √n = √4 = 2.000.
FOLLOW THE METHOD
A reason behind every step.
Formulate null and alternative statistical hypotheses, evaluate Neyman-Pearson decision rules, compute one-sample Z-statistics and Student's t-statistics, calculate exact p-values, and verify the linear scaling of the non-centrality parameter. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 δ.
01
Establish Null and Alternative Hypotheses
Formulate null hypothesis H_0: μ = 0.0 versus upper-tailed alternative H_1: μ > 0.0 with significance level α = 0.05 (critical Z_crit = 1.645).
02
Execute Primary Sampling under Null Hypothesis
Draw 20,000 samples of size n = 4 from N(0, 1). Compute Z = (x̄ - 0)/(1/√4) and verify that Type I error rate P(reject H_0 | H_0 true) = 0.050.
03
Simulate One-Sample Student's t-Test Mode
Switch to unknown variance mode. Compute sample standard deviation s and t-statistic T = x̄ / (s / √4). Verify that critical t_{0.05, 3} = 2.353.
04
Evaluate Non-Central Distribution under Alternative
Inject effect size δ = 1.0 (mean μ = 1.0). Observe that E[Z] shifts to δ √n = 1.0 · 2 = 2.000, and evaluate empirical power 1 - β.
05
Step Standardized Effect Size Across δ = 1.0 to 6.0
Increment effect size across standardized settings δ ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording expected test statistic values E[Z].
06
Conduct Linear Regression on Non-Centrality Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = δ, y = E[Z]). Confirm that empirical slope m = 2.0000 matches theoretical scaling √n = 2.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Single-Sample Z & t Tests mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 2.0000 · x
| # | Standardized Effect Size Magnitude δ (δ) | Expected Test Statistic E[Z] (Statistic E[Z]) | Standardized Effect Size δ | Expected Test Statistic E[Z] | Remove |
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