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 \(\mbox{df} \approx \frac{({s_{1}^2 \over N_1} + {s_2^2 \over N_2})^2}{{s_1^4 \over N_1^2 (N_1 - 1)} + {s_2^4 \over N_2^2 (N_2 - 1)}}\) \(\mbox{se} = \sqrt{\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}}\) \(t = \frac{\mu_1 - \mu_2}{\mbox{se}}\) \(\mbox{se} = \sqrt{\frac{s_1^2}{N_1} + \frac{s_2^2}{N_2}}\) \(\hat{\mu} \pm t_{\alpha \over 2} \mbox{se}\)