##### Using the $C$% confidence interval for $\mu_1 - \mu_2$ to perform two sided two sample $z$ test: full explanation

If 0 is not in the $C$% confidence interval, the difference between the two sample means is significantly different from 0 at significance level $\alpha = 1 - C/100$ If 0 is not in the $C$% confidence interval, the difference between the two sample means is significantly different from 0 at significance level $\alpha = 1 - C/100$. In order to understand why, we have to be aware of the following three points, which are illustrated in the figure above: The lower bound of the $C$% confidence interval for $\mu_1 - \mu_2$ is $(\bar{y}_1 - \bar{y}_2) - z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$, the upper bound is $(\bar{y}_1 - \bar{y}_2) + z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$. That is, we subtract $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ from the difference between the two sample means, and we add $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ to the difference between the two sample means. Here: $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of $\bar{y}_1 - \bar{y}_2$ the critical value $z^*$ is the value under the $z$ distribution with area $C / 100$ between $-z^*$ and $z^*$, and therefore area $\frac{1\,-\,C/100}{2}$ to the right of $z^*$. In the figure above, the red dots represent differences between two sample means, the blue arrows represent the distance $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$. If we set the significance level $\alpha$ at $1 - C/100$, the critical value $z^*$ used for the significance test is the value under the $z$ distribution with area $\frac{\alpha}{2} = \frac{1\,-\,C/100}{2}$ to the right of $z^*$. This is the same critical value $z^*$ as the $z^*$ used for the $C$% confidence interval. We reject the null hypothesis at $\alpha = 1 - C/100$ if the $z$ value $z = \frac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}}$ is at least as extreme as $z^*$. This means that we reject the null hypothesis at $\alpha = 1 - C/100$ if the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is at least $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ removed from 0. In the figure above, the rejection region for $\bar{y}_1 - \bar{y}_2$ is represented by the green area. If 0 is not in the $C$% confidence interval $\Bigg[(\bar{y}_1 - \bar{y}_2) - z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}} \,;\, (\bar{y}_1 - \bar{y}_2) + z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}\Bigg]$, then $\bar{y}_1 - \bar{y}_2$ must be at least $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ removed from 0. In the figure above, the first confidence interval does not contain 0 and thus the difference between the two sample means must be at least $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ removed from 0. The second confidence interval does contain 0, and thus the difference between the two sample means must be less than $z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ removed from 0. In sum, if 0 is not in the $C$% confidence interval $\Bigg[(\bar{y}_1 - \bar{y}_2) - z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}} \,;\, (\bar{y}_1 - \bar{y}_2) + z^* \times \sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}\Bigg]$, we know that the difference between the two sample means is significantly different from 0 at significance level $\alpha = 1 - C/100$.