With independent normal observations and known population SD, the sample mean has normal law with center μ and SE σ / √n at every n. The simulator generates sample means directly as 20 + (σ / √n) × Z, rather than generating n individual observations. Each interval uses its own mean ± z* × SE; the estimator’s interval formula does not use the simulator’s true mean to position its bounds.

A seed-1309 32-bit LCG gives open-unit positions to a Box-Muller normal transform, preparing 2000 standardized draws in 20 batches of 100. Selecting size or spread rescales the same draws; confidence changes margins around unchanged centers. Under this common-draw normal comparison, both mean error and margin scale together, so inclusion stays fixed at the same confidence. Next batch uses another set of standardized draws and may change capture counts. Finite pseudorandom examples do not prove independence or exact coverage.

Confidence 95% describes coverage of this procedure across repeated independent samples under its assumptions, up to numerical normal-critical approximation. Each computed interval either contains the fixed mean or misses it. This is not a posterior probability for that particular interval or a percentage of individual observations within it. Finite batches need not equal confidence exactly, and an all-capture batch does not establish guaranteed coverage. The true mean never moves; sample means move.

All shown intervals and their center markers fit the fixed −40 to 80 window for the prepared batches and allowed parameters; the simulation does not clamp endpoints. Unknown-SD t intervals, non-normal approximations, dependence, biased samples, testing and power are outside scope. Source spread and size affect width, while selected confidence is a separate control. Numeric critical distances invert the Abramowitz–Stegun 26.2.17 normal-tail approximation.