Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalAn interval for $p_1 - p_2$ is read against 0, the value that says the two population proportions are equal. Entirely above 0 gives evidence that $p_1 > p_2$; entirely below gives evidence that $p_1 < p_2$; containing 0 leaves both directions plausible and settles nothing. The checkout interval $(0.018, 0.182)$ for new minus old lies above 0, so the new system's rate is higher by roughly 2 to 18 percentage points. A justification states the interval and what it captures, locates 0, gives the direction, and gives the magnitude, with the order of subtraction named throughout.
Two readings recur. An interval containing 0 gets reported as proof that the groups are equal, though 0 is one plausible value among many and the correct sentence is that the data do not provide convincing evidence of a difference. And the direction gets lost: an interval judged by which side holds more of its length, or a negative interval read as though the first group were ahead, because the order of subtraction was never stated. Reversing that order flips $(-0.14, -0.02)$ into $(0.02, 0.14)$ with no change in meaning, which is exactly why the order has to be written down.
The work
3 ways in · any order
Lesson
Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions
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Reads an interval for a difference against zero: entirely above or below gives a direction, containing zero leaves the question open without establishing equality, and the sign is meaningless until the order of subtraction is stated.
Diagnostic
10-item topic check
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Ten items on differences and zero: intervals containing zero read as proof of equality, direction judged by which side holds more of the interval, negative intervals read as favoring the first group, and causal claims from observed groups. 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.