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

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Where \bar{x}_t is the mean in the total sample and \bar{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{\bar{x}_t-\bar{x}_s}{(1 - w_s)\sqrt{s^2_{\bar{x}_s} + s^2_{\bar{x}_k}}},

where:

p = 2(1-\Phi(|z|)),
s^2_{\bar{x}_k} =\frac{d_{eff} s^2_{x_k}}{n_k},
n_k = n_t - n_g,
s^2_{x_k} = \frac{s^2_{x_t} (n_t - b) - s^2_{x_s} (n_s - b) -  (\bar{x}_t - \bar{x}_s)^2 n_s-  (\bar{x}_t - \bar{x}_k)^2 n_k}{n_k - b},
\bar{x}_k = \frac{\bar{x}_t - w_s\bar{x}_s}{1 - w_s},
s_{\bar{x}_s} is the Standard Error of \bar{x}_s,
b is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise,
s^2_{x_t} and s^2_{x_s} are the estimated standard deviations for the total sample and sub-groups respectively,
s^2_{\bar{x}_t} and s^2_{\bar{x}_s} 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.
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