Carrying Out a Two-Proportion Test
IntroducedA 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.
The test statistic here divides by the pooled standard error, not the un-pooled one.
The two-proportion test uses the pooled SE from H0.
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.
The unrounded standard error gives z ≈ 1.633 and a two-sided p-value ≈ 0.1025. Rounding SE first changes the displayed result.
Two-sided normal tail areas
At |z| = 1.63, the two tails together approximate the two-sided p-value. The central area is its complement.
Using the given rounded SE = 0.061, find (0.30 - 0.20)/0.061 to two decimals.
Using exactly the supplied rounded SE gives 0.10/0.061 ≈ 1.6393, or 1.64. Using the unrounded formula gives 1.633 instead; retain precision in an actual analysis.
Using SE = sqrt(0.25 × 0.75 × (1/100 + 1/100)), calculate z for a difference of 0.10 to three decimal places.
SE ≈ 0.06123724, so z ≈ 1.632993 = 1.633. Retaining precision avoids the 1.64 result from first rounding SE to 0.061.
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.
A non-significant result can still leave practically important differences unresolved. Report uncertainty and the study design.
With a two-sided p-value of 0.10 and α = 0.05, the decision is...
0.10 > 0.05, so fail to reject.
For a symmetric standard-normal z-test with a two-sided alternative, the p-value is...
Use 2P(Z ≥ |z|). Doubling the larger tail gives the wrong answer and may exceed one.
For the worked z ≈ 1.633, match the alternative to its approximate p-value.
Use the prechosen direction. Two-sided normal evidence combines the two equally extreme tails.
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.
Report defined populations, independent random design, count checks, hypotheses, statistic, p-value, decision and contextual conclusion.
A conclusion should be stated in context about the two population proportions.
Always tie the conclusion back to the real groups.
At α = 0.05 the two-sided p-value is approximately 0.1025. Which statements are justified?
The test lacks sufficient evidence at this threshold; it does not establish equality or rule out a meaningful difference.