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

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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}(se^2_s + se^2_k)}},

where:

p = 2\Pr(t_v \ge |t|),
g_k = (g_t-g_s w_s)/(1-w_s),
se_k = \sqrt{\frac{g_k(1 - g_k)}{n_t - n_s - b}},
se_g = \sqrt{\frac{g_g(1 - g_g)}{n_s -b}},
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} } , 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.