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Simulation 5 · Confidence intervals

The Confidence-Interval Dance

"95% confident" does NOT mean there's a 95% chance the true mean is in your interval. It means: if you repeated the study forever, 95% of the intervals you'd build would capture the truth. Watch it happen.

Population: Normal, μ = 50, σ = 10 (σ known — the textbook version). Each row below is one study's interval: x̄ ± z·σ/√n.

n per sample 25
Confidence
Captured μ Missed μ True mean μ = 50
0
Intervals built
—
Captured μ
—
Capture rate
95%
Expected
—
Interval width
Try this:
  1. Draw 100 intervals at 95%. Count the red ones — about 5 in 100 miss, and the researcher who built each red interval has no way to know theirs is the red one. That's the honest meaning of "95% confident."
  2. Switch to 99%: fewer misses, but check the width readout — certainty is purchased with vagueness. Which would you rather report?
  3. Quadruple n from 25 to 100. Width halves (σ/√n again!). Precision comes from sample size, not from wishing.
  4. Turn on Hide μ, reset, draw 1. You're now every real researcher: one interval, no answer key. What can you honestly say?
Sigma says: The parameter μ never moves — it's the intervals that dance. Probability lives in the procedure, not in any single interval. Real studies use t instead of z (σ unknown), but the dance is the same.