Complex Samples Dependent Z-Test - Comparing a Sub-Group Mean to Total

Where ${\overline{x}}_t$ is the mean in the total sample and ${\overline{x}}_s$ is the mean in the sub-group, $n_t$ and $n_s$ are their respective effective sample sizes, and $w_s$ is the proportion of the total population in sub-group $s$:

$z=\frac{{\overline{x}}_t-{\overline{x}}_s}{(1-w_s)\sqrt{s_{{\overline{x}}_s}^2+s_{{\overline{x}}_k}^2}}$,

where:

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

$s_{{\overline{x}}_k}^2=\frac{d_{eff}{s}_{x_k}^2}{n_k}$,

$n_k=n_t-n_g$,

$s_{x_k}^2=\frac{s_{x_t}^2(n_t-b)-s_{x_s}^2(n_s-b)-({\overline{x}}_t-{\overline{x}}_s{)}^2{n}_s-({\overline{x}}_t-{\overline{x}}_k{)}^2{n}_k}{n_k-b}$,

${\overline{x}}_k=\frac{{\overline{x}}_t-w_s{\overline{x}}_s}{1-w_s}$,

$s_{{\overline{x}}_s}$ is the Standard Error of ${\overline{x}}_s$,

$b$ is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise,

$s_{x_t}^2$ and $s_{x_s}^2$ are the estimated standard deviations for the total sample and sub-groups respectively,

$s_{{\overline{x}}_t}^2$ and $s_{{\overline{x}}_s}^2$ are the estimated standard error of the means for the total sample and sub-groups respectively,

$d_{eff}$ is Extra Deff, and

$w_s$ is the proportion of the Population represents by the sub-group.