Which test is equivalent to the 2-proportion z-test when comparing two independent proportions with large samples?

Which test is equivalent to the 2-proportion z-test when comparing two independent proportions with large samples?

• A. McNemar's test
• B. Chi-square test of homogeneity with 1 degree of freedom
• C. Fisher's exact test
• D. Paired t-test

Answer: (B)

When comparing two independent proportions with large samples, the difference between the sample proportions is approximately normally distributed, which is why the two-proportion z-test works. This same comparison can be viewed as a test on a 2x2 contingency table using the Pearson chi-square statistic. The chi-square test of homogeneity with 1 degree of freedom effectively asks whether the distribution of outcomes is the same across the two groups, which is exactly what the two-proportion z-test assesses. In this 2x2 setup, the chi-square statistic and the z statistic are linked: chi-square with 1 degree of freedom equals z squared, so they yield the same p-value in large samples. That makes the chi-square test of homogeneity with 1 df equivalent to the two-proportion z-test. The other options don’t fit this scenario as neatly: McNemar's test is for paired (related) data, not two independent groups; Fisher's exact test is an exact method for small counts and isn’t the standard large-sample equivalence; the paired t-test is for comparing means of paired continuous data, not proportions.

When comparing two independent proportions with large samples, the difference between the sample proportions is approximately normally distributed, which is why the two-proportion z-test works. This same comparison can be viewed as a test on a 2x2 contingency table using the Pearson chi-square statistic. The chi-square test of homogeneity with 1 degree of freedom effectively asks whether the distribution of outcomes is the same across the two groups, which is exactly what the two-proportion z-test assesses.

In this 2x2 setup, the chi-square statistic and the z statistic are linked: chi-square with 1 degree of freedom equals z squared, so they yield the same p-value in large samples. That makes the chi-square test of homogeneity with 1 df equivalent to the two-proportion z-test.

The other options don’t fit this scenario as neatly: McNemar's test is for paired (related) data, not two independent groups; Fisher's exact test is an exact method for small counts and isn’t the standard large-sample equivalence; the paired t-test is for comparing means of paired continuous data, not proportions.