Carrying Out a Two-Proportion Test

A ten-point sample difference is not yet a conclusion

• For 30/100 versus 20/100, the observed difference is 0.10 and the pooled null standard error is about 0.06124.
• A two-sided equality test asks whether this difference, or one more extreme in either direction, is unusual under the checked null model.

Calculate before rounding

• Use $z=(\hat p_1-\hat p_2-0)/SE_0$. Substituting the unrounded null standard error gives $z\approx1.633$.
• Rounding $SE_0$ early to 0.061 changes the displayed statistic to about 1.64. Keep extra digits until the final result.

Use both tails of the null distribution

• For the two-sided alternative, the p-value is $P(Z\le-1.633)+P(Z\ge1.633)\approx0.1025$.
• Equivalently, double the smaller tail. Do not take only the left area up to the positive statistic; that is about 0.9488.

Apply the chosen significance level

• At $\alpha=0.05$, the p-value is larger than the threshold, so fail to reject $H_0:p_1=p_2$.
• The data do not give sufficient evidence of a difference in the population on-time rates at this level. They do not prove equal rates.

Separate sample size from practical size

• The observed gap is ten percentage points, but the test has substantial uncertainty. An estimated difference can be important even without a rejection.
• An interval can help describe plausible effect sizes, using its own unpooled formula. Do not assert universal numerical agreement between that interval and this test.

Check every link in the report

• Report defined populations, independent random design, count checks, hypotheses, statistic, p-value, decision and contextual conclusion.
• For a one-sided question, use the tail selected by that alternative. Changing the question after observing the data can create misleading evidence.

Write from a statistic beyond 1.96

• A two-sided two-proportion test has z equal to 2.1.
• The 5 percent critical values are about negative 1.96 and 1.96. The lesson statistics 1.64 and 1.633 are another test.