# Statistical Inference Formulas

Since I could not find a list of formulas anywhere, I thought I would create one.

### Single Population

Test df Test se Test Statistic Confidence se Confidence interval
$\mu$ when $\sigma$ known - $\mbox{se} = \frac{\sigma}{\sqrt{n}}$ $z = \frac{\hat{\mu}}{\mbox{se}}$ $\mbox{se} = \frac{\sigma}{\sqrt{n}}$ $\hat{\mu} \pm z_{\alpha \over 2} \mbox{se}$
$\mu$ when $\sigma$ unknown $\mbox{df} = N-1$ $\mbox{se} = \frac{s}{\sqrt{N}}$ $t = \frac{\hat{\mu}}{\mbox{se}}$ $\mbox{se} = \frac{s}{\sqrt{N}}$ $\hat{\mu} \pm t_{\alpha \over 2} \mbox{se}$
Population proportion $p$ - $\mbox{se} = \sqrt{\frac{p_0 (1 - p_0)}{n}}$ $z = \frac{\hat{p} - p_0}{\mbox{se}}$ $\mbox{se} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$ $\hat{p} \pm z_{\alpha \over 2} \mbox{se}$

### Two Populations

Test df Test se Test Statistic Confidence se Confidence interval
two means unequal variances $\mbox{df} \approx \frac{({s_{1}^2 \over N_1} + {s_2^2 \over N_2})^2}{{s_1^4 \over N_1^2 (N_1 - 1)} + {s_2^4 \over N_2^2 (N_2 - 1)}}$ $\mbox{se} = \sqrt{\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}}$ $t = \frac{\mu_1 - \mu_2}{\mbox{se}}$ $\mbox{se} = \sqrt{\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}}$ $\hat{\mu} \pm t_{\alpha \over 2} \mbox{se}$