IntroStatsCalc

Two-Proportion Z-Test

Compare two population proportions (2-PropZTest and 2-PropZInt).

Inputs

Sample 1

Sample 2

Alternative hypothesis Ha
Significance level α
Confidence level

Results update as you type.

Results

Fail to reject H₀: z = 1.8463, p-value = 0.0649

The p-value (0.0649) is greater than α = 0.05, so the data do not give enough evidence to conclude that the proportion in population 1 is different from the proportion in population 2.

Standard normal curve. The shaded p-value region is both tails beyond z = 1.85, with area 0.0649. Dashed lines mark the critical values −1.96 and 1.96.−4−2024z = 1.85critical −1.96critical 1.96

Results

p̂10.4
p̂20.288
Pooled p̂0.3429
Test statistic z1.8463
p-value0.0649
95% CI for p1 − p2(−0.0063, 0.2303)
Decision at α = 0.05Fail to reject H₀

On the TI-84

STAT → TESTS → 6:2-PropZTest.

2-PropZTest 48, 120, 36, 125, ≠

Interval: STAT → TESTS → B:2-PropZInt.

2-PropZInt 48, 120, 36, 125, 0.95

Show steps
  1. Hypotheses. H₀: p1 = p2   Ha: p1 ≠ p2
  2. Sample proportions. p̂1 = 48/120 = 0.4, p̂2 = 36/125 = 0.288. Pooled: p̂ = (48 + 36) ÷ (120 + 125) = 0.3429.
  3. Test statistic. z = (p̂1 − p̂2) ÷ √(p̂(1 − p̂)(1/n1 + 1/n2)) = 0.112 ÷ 0.0607 = 1.8463
  4. p-value. Two-tailed test, so double the tail area beyond the test statistic: p-value = 0.0649. Since 0.0649 > 0.05, fail to reject H₀.
  5. Critical-value method. At α = 0.05, the rejection region is z ≤ −1.96 or z ≥ 1.96. The test statistic z = 1.8463 is not in the rejection region, so we fail to reject H₀. (Same decision as the p-value method.)
  6. Confidence interval (unpooled SE = 0.0603): 0.112 ± 1.96 · 0.0603 → (−0.0063, 0.2303).
  7. Conclusion. The p-value (0.0649) is greater than α = 0.05, so the data do not give enough evidence to conclude that the proportion in population 1 is different from the proportion in population 2.

About this calculator

When to use this

Use a two-proportion z-test to compare percents from two independent groups, such as the pass rate in two course formats. The test uses the pooled proportion because H0 assumes the two population proportions are equal; the confidence interval uses each sample's own proportion.

Each sample should have at least 5 successes and 5 failures (some textbooks use 10).

More inference calculators: CI for a Mean · CI for a Proportion · Sample Size · Z-Test (mean) · T-Test (mean) · 1-Prop Z-Test · 2-Sample T-Test · Paired T-Test