If we want to perform a two-tail test for differences between two populations of nominal data with exactly two categories, we can employ either the z-test of $p _ { 1 } - p _ { 2 }$ , or the chi-squared test of a contingency table. (Squaring the value of the z-statistic yields the value of the $x ^ { 2 }$ -statistic.)

Question

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Both the z-test for $p_1 - p_2$ and the chi-squared test for a contingency table can be used to test differences between two proportions in nominal data with two categories. Squaring the z-statistic indeed yields the chi-squared statistic in this context, showing a mathematical relationship between the two tests.

If We Want to Perform a Two-Tail Test for Differences p1−p2p _ { 1 } - p _ { 2 }p1​−p2​

If We Want to Perform a Two-Tail Test for Differences $p _ { 1 } - p _ { 2 }$