Computational Details

Under the sample size assumptions, the sampling distribution of the difference in the sample means,

$$\overline{x}_1-\overline{x}_2$$

is normally distributed with mean $\mu_1-\mu_2$ and standard deviation

$$\sigma_{\overline{x}_1-\overline{x}_2}\;=\;\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$$

When the means of the two populations are equal,

$$\mu_1\;=\;\mu_2,$$

We may calculate the standardized or $Z$ -score for the difference between the sample means as:

$$Z\;=\;\frac{\overline{x}_1-\overline{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$$

(This value is in cell B12)