Among a random sample of college-age students, 6% of the 473 men said they had been adopted, compared to 4% of the 552 women. Which test is appropriate to determine if there is a significant difference between adoption rates by gender?

Among a random sample of college-age students, 6% of the 473 men said they had been adopted, compared to 4% of the 552 women. Which test is appropriate to determine if there is a significant difference between adoption rates by gender?

• A. Chi-square test of independence
• B. ANOVA
• C. 2-proportion z-test
• D. Paired t-test

Answer: (C)

When you want to know if two independent groups differ on a binary outcome, compare their proportions with a two-proportion z-test. Here, adoption is a yes/no outcome in two independent samples (men and women), so this test directly assesses whether the difference in adoption rates is statistically significant. You can treat the data as p1 = 0.06 from n1 = 473 and p2 = 0.04 from n2 = 552. Under the null hypothesis that the true rates are equal, use the pooled proportion p̂ = (0.06×473 + 0.04×552) / (473 + 552) ≈ 0.0493. The standard error for the difference in proportions is sqrt[p̂(1 − p̂)(1/n1 + 1/n2)] ≈ sqrt(0.0493×0.9507×(1/473 + 1/552)) ≈ 0.0136. The observed difference is 0.06 − 0.04 = 0.02, giving a z ≈ 0.02 / 0.0136 ≈ 1.48, which corresponds to a two-sided p-value around 0.14—not significant at common alpha levels. The two-proportion z-test is the best fit here because it targets the specific question: do the adoption rates differ by gender in two independent samples with a binary outcome? A chi-square test of independence could also apply to a 2×2 table, but the two-proportion z-test is the direct, most straightforward approach for this comparison. Paired t-tests and ANOVA are for different data types (paired or continuous outcomes).

When you want to know if two independent groups differ on a binary outcome, compare their proportions with a two-proportion z-test. Here, adoption is a yes/no outcome in two independent samples (men and women), so this test directly assesses whether the difference in adoption rates is statistically significant.

You can treat the data as p1 = 0.06 from n1 = 473 and p2 = 0.04 from n2 = 552. Under the null hypothesis that the true rates are equal, use the pooled proportion p̂ = (0.06×473 + 0.04×552) / (473 + 552) ≈ 0.0493. The standard error for the difference in proportions is sqrt[p̂(1 − p̂)(1/n1 + 1/n2)] ≈ sqrt(0.0493×0.9507×(1/473 + 1/552)) ≈ 0.0136. The observed difference is 0.06 − 0.04 = 0.02, giving a z ≈ 0.02 / 0.0136 ≈ 1.48, which corresponds to a two-sided p-value around 0.14—not significant at common alpha levels.

The two-proportion z-test is the best fit here because it targets the specific question: do the adoption rates differ by gender in two independent samples with a binary outcome? A chi-square test of independence could also apply to a 2×2 table, but the two-proportion z-test is the direct, most straightforward approach for this comparison. Paired t-tests and ANOVA are for different data types (paired or continuous outcomes).