In Testing Where $p _ { 1 } \text { and } p _ { 2 }$

Essay

In testing $H _ { 0 } : p _ { 1 } - p _ { 2 } = 0 \text { versus } H _ { \text {at } } : p _ { 1 } - p _ { 2 } \neq 0 \text { where } p _ { 1 } \text { and } p _ { 2 }$ where $p _ { 1 } \text { and } p _ { 2 }$ denote the two population proportions, the standardized variable $z = \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) / \sqrt { \hat { p } \hat { q } \left( \frac { l } { m } + \frac { l } { n } \right) } , \text { where } \hat { p }$ is an estimate of the common value of $p _ { 1 } \text { and } p _ { 2 }$ and m and n are the two sample sizes, has approximately a standard normal distribution when __________.