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Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means

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An interval for $\mu_1 - \mu_2$ is read against 0. Entirely above gives evidence that $\mu_1 > \mu_2$, entirely below the reverse, and containing 0 settles nothing about direction. The teaching-methods interval $(2.40, 12.00)$ for A minus B lies above 0, so method A's mean runs higher by roughly 2.4 to 12.0 points. The width comes from the confidence level, the two groups' variability, and the two sample sizes, and it answers a second question: how large the plausible differences are.

An interval containing 0 gets reported as showing the two means are equal, though a 4-point gap can sit comfortably inside $(-1.2, 4.8)$ and the correct sentence is that no difference was detected. Direction gets judged by which side holds more of the interval, or read off the sign with no order of subtraction stated. Width gets read as importance, when a precise $(0.2, 0.6)$ on a 100-point exam is a real and trivial difference and a wide $(1, 40)$ is a real and unpinned one. And causation gets asserted from two intact groups that were never randomly assigned.

A minus B, three possible outcomes 0 (2.40, 12.00): evidence that A is higher 0 (-1.2, 4.8): no convincing evidence of a difference and NOT evidence that the means are equal: 4.8 is inside 0 (-9.4, -2.1): evidence that B is higher
Only the middle case is routinely misreported. It leaves differences in both directions plausible, which is a failure to decide rather than a finding that the two means match.
both exclude 0, and they say very different things 0 (0.2, 0.6) on a 100-point exam real, precisely pinned, and too small to act on 0 (1, 40): a difference exists, size barely pinned down report both answers: IS there a difference, and HOW BIG
Excluding zero settles only the first question. The endpoints settle the second, and a report that gives the verdict without the range has answered half of what the interval knows.

The work

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Lesson
Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means ›

Reads an interval for a difference of means against zero, refuses the reading that turns a straddling interval into a finding of equality, separates the width's precision from the difference's importance, and writes a four-part justification with its hedge intact.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on differences of means and zero: intervals containing zero read as proof of equality, direction judged by which side holds more, signs read without the subtraction order, and narrow intervals read as large effects. Take it cold to find your habit, or after the lesson to check it is gone.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.

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