Paired Z-Test of Means

Where ${\overline{x}}_1$ and ${\overline{x}}_2$ are the two means, the test statistic is:

$z=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_{{\overline{x}}_1-{\overline{x}}_2}}$

where:

$p\approx 2(1-\mathrm{\Phi }(|z|))$,

$s_{{\overline{x}}_1-{\overline{x}}_2}$ is computed as the Standard Error of $d$,

the value for the $i$th observation is computed as $d_i=x_{i1}-x_{i2}$, and

$x_1,x_2$ are the observed values on the two variables.