This simulator sets an observed sample mean against a normal null model with known standard deviation. It converts the result to a z statistic, shades the two-sided tails that make up the p-value, draws the decision boundaries for your chosen significance level, and can simulate many repeated tests where the null is actually true.
• A real-time 3D scene with 4 inspectable parts (Standard normal null distribution, Two-sided p-value tails, Decision boundaries and Repeated-null test counter), with home view, focus-selected-part, auto-rotate, expand, instrument-cover and hide-labels scene tools, plus a model response curve beneath the scene. • Experiment controls: Observed sample mean (-1-1); Sample size (5-100); Known population SD (0.5-2); Significance level (1%, 5%, 10%); Repeatable random seed (1-99); show animated explanatory markers; pause/resume, 0.1 s and 1 s single-step buttons, four playback speeds and a restart button. • A Curves & measurements tab with a parameter-comparison chart, a live-measurements chart, the model equations and snapshot readouts (Observed z statistic; Two-sided p-value; Reject H₀ (1=yes); Positive critical z; Null simulations; Observed null rejection fraction). • An Experiments tab with 2 guided presets (mean at null and small effect, larger sample) and a Model verification bench that runs independent fresh models, plus a timestamped event log and a copyable trial report. • A Learn & assess tab with guided lessons, a knowledge-check quiz with reset and a written model-scope statement linking to a technical reference.
The z statistic is the observed mean divided by sigma over the square root of n, and the two-sided p-value is 2 times one minus the standard normal CDF of the absolute z. You reject the null when p is below alpha, equivalently when the absolute z exceeds the critical value, which the lab shows for 1%, 5% and 10% significance.
Changing the sample size or the known standard deviation changes the standard error, so the same observed mean can move from non-significant to significant without any change in the underlying effect.
A p-value is computed assuming the null is true; it is not the probability that the null is true, nor the chance that a particular rejection is wrong. The repeated-null counter draws independent z values under the null and reports the fraction rejected, which should hover near alpha - the long-run Type I error rate - with finite-run fluctuation.
This is a known-variance z test, not a t test with an estimated variance, and null simulations stop after 500 trials. The normal-CDF approximation is accurate to about one part in ten million.
No. The p-value is the probability of a result at least as extreme as observed, computed assuming the null is true. It conditions on the null rather than assigning the null a probability.
Under repeated tests where the null is true, alpha is the long-run fraction of tests that will wrongly reject - the Type I error rate. The repeated-null counter shows that fraction wobbling around alpha over a finite number of trials.
Because the standard error shrinks as the sample size grows, so the same small mean difference gives a larger z statistic and a smaller p-value. Significance reflects effect size relative to precision.
No. The lab assumes the population standard deviation is known, which gives an exact z test. A t test with an estimated variance has heavier tails and is not modeled here.