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

Where [math]\displaystyle{ \bar{x}_t }[/math] is the mean in the total sample and [math]\displaystyle{ \bar{x}_s }[/math] is the mean 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{ z=\frac{\bar{x}_t-\bar{x}_s}{(1 - w_s)\sqrt{s^2_{\bar{x}_s} + s^2_{\bar{x}_k}}} }[/math],

where:

[math]\displaystyle{ p = 2(1-\Phi(|z|)) }[/math],

[math]\displaystyle{ s^2_{\bar{x}_k} =\frac{d_{eff} s^2_{x_k}}{n_k} }[/math],

[math]\displaystyle{ n_k = n_t - n_g }[/math],

[math]\displaystyle{ 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} }[/math],

[math]\displaystyle{ \bar{x}_k = \frac{\bar{x}_t - w_s\bar{x}_s}{1 - w_s} }[/math],

[math]\displaystyle{ s_{\bar{x}_s} }[/math] is the Standard Error of [math]\displaystyle{ \bar{x}_s }[/math],

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

[math]\displaystyle{ s^2_{x_t} }[/math] and [math]\displaystyle{ s^2_{x_s} }[/math] are the estimated standard deviations for the total sample and sub-groups respectively,

[math]\displaystyle{ s^2_{\bar{x}_t} }[/math] and [math]\displaystyle{ s^2_{\bar{x}_s} }[/math] are the estimated standard error of the means for the total sample and sub-groups respectively,

[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.