Independent Samples Z-Test - Comparing Two Means with Unequal Variances

The test statistic is:

$z = \frac{\bar{x} - \bar{y}}{\sqrt{\frac{\sigma_x}{m} + \frac{\sigma_y}{n}}}$

where:

$\bar{x}$ and $\bar{x}$ are the average values of variables $x$ and $y$ respectively, where each of these variables represents the data from two independent groups,

the groups have sample sizes of $m$ and $n$ respectively,

$\sigma^2_x$ and $\sigma^2_y$ are the variances in the two groups,

$p = 2(1-\Phi(|z|))$