Complex Samples Dependent T-Test - Comparing a Sub-Group Proportion to Total

Where $g_t$ is the proportion in the total sample and $g_s$ is the proportion 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 $s$:

$t=\frac{g_t-g_s}{(1-w_s)\sqrt{d_{eff}(s{e}_s^2+s{e}_k^2)}}$,

where:

$p=2\mathrm{Pr}(t_v\ge |t|)$,

$g_k=(g_t-g_s{w}_s)/(1-w_s)$,

$s{e}_k=\sqrt{\frac{g_k(1-g_k)}{n_t-n_s-b}}$,

$s{e}_g=\sqrt{\frac{g_g(1-g_g)}{n_s-b}}$,

$v=\frac{(\frac{s{e}_s^2}{n_s}+\frac{s{e}_k^2}{n_k}{)}^2}{\frac{(\frac{s{e}_s^2}{n_s}{)}^2}{n_s-b}+\frac{(\frac{s{e}_k^2}{n_k}{)}^2}{n_k-b}}$, or, if Weights and significance is set to Un-weighted sample size in tests (see Weights, Effective Sample Size and Design Effects), $v=n_t-2b$

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

$d_{eff}$ is Extra Deff, and

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