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

The test statistic is:

[math]\displaystyle{ z = \frac{\bar{x} - \bar{y}}{\sqrt{\frac{\sigma^2_x}{m} + \frac{\sigma^2_y}{n}}} }[/math]

where:

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

the groups have sample sizes of [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] respectively,

[math]\displaystyle{ \sigma^2_x }[/math] and [math]\displaystyle{ \sigma^2_y }[/math] are the variances in the two groups,

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