quartz/content/Obsidian Vault/joint distribution.md

#stats

variance: \frac{\sigma_m^2}{n_m} + \frac{\sigma_w^2}{n_w}

hypothesis test: $H_0: (\mu_m-\mu_w)=0$ $H_a: (\mu_m-\mu_w) < 0$ \alpha = 0.05

\frac{(\bar X_m-\bar X_w)-(\mu_m-\mu_w)}{\sqrt{\frac{\sigma_m^2}{n_m} + \frac{\sigma_w^2}{n_w}}}

If you need to estimate it \frac{(\bar x_m-\bar x_w)-(\mu_m-\mu_w)}{\sqrt{\frac{s_m^2}{n_m} + \frac{s_w^2}{n_w}}}