Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA claimed value inside an interval is plausible and one outside is contradicted. The battery interval $(120.8, 128.4)$ contains the manufacturer's claim of 128, so the data do not contradict it even though $\bar{x} = 124.6$ sits below. For a mean difference, 0 is the no-change value: the paired interval $3.6 \pm 3.43 = (0.17, 7.03)$ lies entirely above 0, giving evidence of a mean improvement of roughly 0.2 to 7.0 points. The margin of error $t^{*}\frac{s}{\sqrt{n}}$ covers sampling variability only, and quadrupling $n$ halves it.
An interval containing 0 gets reported as proof of no effect, when 0 is one plausible value among many and the correct sentence is that no change was detected. The sign of a difference interval gets read without the order of subtraction, so nobody can tell which direction it points. The margin of error gets credited with covering bad question wording or nonresponse, which no term in the formula touches. And a narrower interval bought by lowering the confidence level gets described as more precision, when the method now captures the parameter less often.
The work
3 ways in · any order
Lesson
Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference
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Turns an interval for a mean into a verdict on a claim, makes zero the decisive value for a mean difference, separates what the margin of error covers from the nonsampling errors it cannot see, and writes a four-part justification with its hedge intact.
Diagnostic
10-item topic check
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Ten items on judging claims about means: intervals containing zero read as proof of no change, signs read without the subtraction order, margins of error credited with covering bad questions, and lower confidence sold as more precision. Take it cold to find your habit, or after the lesson to check it is gone.
Targeted Practice
Drill a single misconception
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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.