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Justifying a Claim Based on a Confidence Interval for a Population Proportion

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A confidence interval is the set of plausible values for the parameter, so a claimed value inside it is plausible and a claimed value outside it is contradicted by the data. Directional claims are settled by position: an interval lying entirely above a threshold supports the claim that $p$ exceeds it, an interval entirely below supports the reverse, and an interval straddling the threshold supports neither. A justification states the interval and what it estimates, locates the claimed value relative to it, and concludes about the parameter in context with hedged language.

The logic gets run backwards in both directions. A value inside the interval is treated as rejected, or as proven, when it has only failed to be excluded and shares that status with every other value in the range. A value outside is treated as disproven, though a 95% method misses about one interval in twenty. And a straddling interval gets forced into a verdict, either by reading only its upper end as evidence of a majority or by converting the undecided result into a positive claim that no difference exists. The confidence level compounds it when read as the probability that this one interval is correct rather than as the method's long-run capture rate.

claim under test: p = 0.50 0.50 0.572 0.668 entirely above: evidence AGAINST p = 0.50, and support for a majority 0.50 0.47 0.55 contains 0.50: plausible, and so is every other value in the range
Position decides the verdict. The second interval does not select 0.50: it fails to exclude it, along with 0.49 and 0.54, which is why no directional claim survives.
claimed value correct conclusion the backwards reads INSIDE plausible: not ruled out by these data "so the claim is proven" "so we reject the claim" OUTSIDE evidence against it "so the claim is disproven" a 95% method misses about 1 interval in 20, which is why nothing here is proof
Both rows keep their hedge. The interval reports which values a sample can and cannot rule out, and a method that misses one time in twenty cannot deliver proof in either direction.

The work

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Lesson
Justifying a Claim Based on a Confidence Interval for a Population Proportion ›

Turns an interval into a verdict on a claim: inside means plausible and outside means contradicted, with directional claims settled by whether the whole interval clears the threshold, and a three-part justification that names the population and keeps its hedge.

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

Ten items on judging claims with intervals: values rejected for being inside, claims declared proven, straddling intervals forced into a direction, and confidence read as the probability this one interval is right. 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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