Zero Inside Means Nothing Was Shown, Not That Nothing Happened

You'll learnhow to turn an interval for a mean into a verdict on a claimed value, why zero is the decisive value for a mean difference and what containing it does and does not establish, what the margin of error covers and cannot cover, and how to write the four-part justification.

A manufacturer claims a mean lifetime of 128 hours, the sample mean comes out at 124.6, and the data still do not contradict the claim — because the interval (120.8, 128.4) reaches past it, and a verdict is a membership test, not a comparison of two point values. This page runs every such test in code. For a mean difference the decisive member is zero, the no-change value: a paired study's interval (0.17, 7.03) clears it and shows improvement; the hypothetical (−1.2, 4.8) straddles it, and the correct sentence is that no change was detected — not that none occurred, since a 4.8-point gain sits inside that interval too. Then the margin of error is taken apart lever by lever — confidence, sample size, variability, each move priced by integration — and the two habits it cannot survive are executed: crediting it with covering a leading question, and buying a narrower interval at 80 percent confidence as if the missing capture rate were free.

8 STEPS · 6 QUICK CHECKS · PLAUSIBLE VALUES · ZERO AND MEAN DIFFERENCES · MARGIN OF ERROR · v1

inside = plausible · outside = contradicted · zero is the no-change value · MoE covers sampling only

Before you start

What you're looking at

Number lines with confidence intervals as brackets, and claimed values as dashed vertical lines. Every verdict is a membership test the page computes.

The question

When does an interval contradict a claim? What does zero inside a difference interval actually establish? And which failures can the margin of error never see?

Watch for

Step 2: the sample mean lands below the claim and the claim survives anyway. Position relative to the endpoints is the whole criterion.

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