Paired Z-Test of Means

Where [math]\displaystyle{ \bar x_1 }[/math] and [math]\displaystyle{ \bar x_2 }[/math] are the two means, the test statistic is:

[math]\displaystyle{ z=\frac{\bar x_1-\bar x_2}{s_{{\bar x_1} - {\bar x_2}}} }[/math]

where:

[math]\displaystyle{ p \approx 2(1-\Phi(|z|)) }[/math],

[math]\displaystyle{ s_{{\bar x_1} - {\bar x_2}} }[/math] is computed as the Standard Error of [math]\displaystyle{ d }[/math],

the value for the [math]\displaystyle{ i }[/math]th observation is computed as [math]\displaystyle{ d_i = x_{i1} - x_{i2} }[/math], and

[math]\displaystyle{ x_1,x_2 }[/math] are the observed values on the two variables.