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

Where [math]\displaystyle{ g_t }[/math] is the proportion in the total sample and [math]\displaystyle{ g_s }[/math] is the proportion in the sub-group, [math]\displaystyle{ n_t }[/math] and [math]\displaystyle{ n_s }[/math] are their respective effective sample sizes, and [math]\displaystyle{ w_s }[/math] is the proportion of the total population in sub-group [math]\displaystyle{ s }[/math]:

[math]\displaystyle{ t=\frac{g_t - g_s}{(1 - w_s)\sqrt{d_{eff}(se^2_s + se^2_k)}} }[/math],

where:

[math]\displaystyle{ p = 2\Pr(t_v \ge |t|) }[/math],

[math]\displaystyle{ g_k = (g_t-g_s w_s)/(1-w_s) }[/math],

[math]\displaystyle{ se_k = \sqrt{\frac{g_k(1 - g_k)}{n_t - n_s - b}} }[/math],

[math]\displaystyle{ se_g = \sqrt{\frac{g_g(1 - g_g)}{n_s -b}} }[/math],

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

[math]\displaystyle{ b }[/math] is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise,

[math]\displaystyle{ d_{eff} }[/math] is Extra Deff, and

[math]\displaystyle{ w_s }[/math] is the proportion of the Population represents by the sub-group.