Carrying Out a Test for the Difference Between Two Population Proportions
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalThe two-proportion test statistic is $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$, with nothing subtracted in the numerator because the null puts the difference at 0. For 138 of 300 against 90 of 250: $z = \frac{0.10}{0.0422} \approx 2.37$, one-sided p-value about 0.0089 and two-sided about 0.0178, so at $\alpha = 0.05$ the null is rejected. The conclusion states the comparison, the decision, the direction, and both populations, with a causal verb only if subjects were randomly assigned.
A large p-value gets converted into a finding of equality: the two systems perform equally well, there is no difference, the new one is no better. All three accept the null, and with modest samples a real gap of 5 or 8 percentage points can easily go undetected, so the correct report is that there is not convincing evidence of a difference. The other recurring slips are mechanical: the unpooled standard error used in the test, the observed difference subtracted again in the numerator, and a conclusion that gives no direction because the order of subtraction was never stated.
The work
3 ways in · any order
Lesson
Carrying Out a Test for the Difference Between Two Population Proportions
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Runs the two-proportion z-test end to end with the pooled standard error, writes a conclusion carrying the direction and both populations, refuses the three sentences that turn a large p-value into equality, and lines the test up against the interval.
Diagnostic
10-item topic check
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Ten items on carrying out a two-proportion test: equality concluded from a large p-value, conclusions without a direction, unpooled standard errors in the statistic, and causal verbs 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.