📏 Confidence Intervals, the Z-Test, and Coverage

About this simulation

Every time you draw a fresh random sample from the population, you compute a sample mean and build a 95% (or whatever level you choose) confidence interval around it. This simulation draws many such samples, one after another, and stacks the resulting intervals as horizontal bars along a depth axis. A green bar means the interval happened to capture the true population mean μ (the red plane); a red bar means it missed.

How it works

• Coverage is a property of the procedure, not of any single interval — over many repeated samples, roughly C% of the intervals built this way will capture μ, where C is your chosen confidence level.
• Every interval is built from the same sampling distribution — the bell-shaped curve of possible sample means, with spread equal to the standard error σ/√n, guaranteed to be approximately normal by the Central Limit Theorem regardless of the population's own shape.
• The Z-test reuses this exact machinery: it standardizes the latest sample mean against a hypothesized μ₀, producing Z = (x̄ − μ₀) / SE. Large |Z| means the sample mean sits far out in the tails of the sampling distribution if μ₀ were true — that's a small p-value.
• Narrower intervals (smaller confidence, larger n) miss more often; wider intervals (higher confidence, smaller n) waste less certainty but miss less often, by design.

Controls

• Sample size n — how many observations go into each drawn sample; larger n shrinks the standard error and the bell curve.
• Confidence level — sets the critical z-value and interval half-width for the confidence intervals.
• Population σ — the true population standard deviation, controlling how spread out individual observations are.
• Hypothesized μ₀ — the value tested by the Z-test; moves the yellow test-plane and updates the Z-statistic and p-value for the latest sample.
• 🎲 Draw one sample / ▶ Auto-draw ×20 / ↺ Reset — control how intervals accumulate. Drag to rotate, scroll to zoom.

Did you know?

A 95% confidence interval does not mean "there's a 95% chance μ is in this particular interval." Once computed, an interval either contains μ or it doesn't — the 95% describes how the interval-building procedure behaves if you repeated it forever, which is exactly what this simulation lets you watch happen.