| Response | Explanatory | Numerical_Quantity | Parameter | Statistic |
|---|---|---|---|---|
| quantitative |
|
mean | \(\mu\) | \(\bar{x}\) |
| quantitative |
|
standard deviation | \(\sigma\) | \(s\) |
| binary categorical |
|
proportion | \(p\) | \(\hat{p}\) |
| multi-level categorical |
|
Chi-square statistic |
|
\(\chi^2\) |
| quantitative | binary categorical | difference in means | \(\mu_1 - \mu_2\) | \(\bar{x}_1 - \bar{x}_2\) |
| binary categorical | binary categorical | difference in proportions | \(p_1 - p_2\) | \(\hat{p}_1 - \hat{p}_2\) |
| quantitative | quantitative | correlation | \(\rho\) | \(R\) |
| quantitative | multli-level categorical | F statistic |
|
\(F\) |
| multli-level categorical | multli-level categorical | Chi-square statistic |
|
\(\chi^2\) |
| Response | Explanatory | Numerical_Quantity | Test_Statistic | Distribution | Assumptions |
|---|---|---|---|---|---|
| quantitative |
|
mean | \(\frac{\bar{x} - \mu_o}{s/\sqrt{n}}\) | \(t(df = n - 1)\) | \(n \geq 30\) or data are normal |
| binary categorical |
|
proportion | \(\frac{\hat{p} - p_o}{\sqrt{\frac{p_o(1 - p_o)}{n}}}\) | \(N(0, 1)\) | Ten successes, Ten failures |
| multi-level categorical |
|
Chi-square statistic | \(\sum \frac{(\textrm{Obs.} - \textrm{Exp.})^2}{\textrm{Exp.}}\) | \(\chi^2(df = (k-1))\) | All Exp. \(\geq 5\) |
| quantitative | binary categorical | difference in means | \(\frac{\bar{x}_1 - \bar{x}_2 - 0}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}}\) | \(t(df = \min(n_1, n_2) - 1)\) | \(n_1, n_2 \geq 30\) or data are normal; independent samples |
| binary categorical | binary categorical | difference in proportions | \(\frac{\hat{p}_1 - \hat{p}_2 - 0}{\sqrt{\frac{\hat{p}(1 - \hat{p})}{n_1} + \frac{\hat{p}(1 - \hat{p})}{n_2}}}\) | \(N(0, 1)\) | Ten successes, Ten failures in each category; independent samples |
| quantitative | quantitative | correlation | \(\frac{R - 0}{\sqrt{\frac{1 - R^2}{n - 2}}}\) | \(t(df = n - 2)\) | \(n \geq 30\) or data are normal |
| quantitative | multli-level categorical | F statistic | \(\frac{\textrm{MSG}}{\textrm{MSE}} = \frac{n-k}{k-1} \frac{\sum n_i (\bar{x}_i - \bar{x})^2}{\sum (x - \bar{x}_i)^2}\) | \(F(df_1 = k-1, df_2 = n-k)\) | Within each group, data are Normal with equal variance |
| multli-level categorical | multli-level categorical | Chi-square statistic | \(\sum \frac{(\textrm{Obs.} - \textrm{Exp.})^2}{\textrm{Exp.}}\) | \(\chi^2(df = (k_1-1)\cdot (k_2 - 1))\) | All Exp. \(\geq 5\) |
| Response | Explanatory | Numerical_Quantity | Confidence_Interval | Distribution | Assumptions |
|---|---|---|---|---|---|
| quantitative |
|
mean | \(\bar{x} \pm t^*s/\sqrt{n}\) | \(t(df = n - 1)\) | \(n \geq 30\) or data are normal |
| binary categorical |
|
proportion | \(\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\) | \(N(0, 1)\) | Ten successes, Ten failures |
| quantitative | binary categorical | difference in means | \(\bar{x}_1 - \bar{x}_2 \pm t^* \sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}\) | \(t(df = \min(n_1, n_2) - 1)\) | \(n_1, n_2 \geq 30\) or data are normal |
| binary categorical | binary categorical | difference in proportions | \(\hat{p}_1 - \hat{p}_2 \pm z^* \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}\) | \(N(0, 1)\) | Ten successes, Ten failures in each category |
| quantitative | quantitative | correlation | \(R \pm t^* \sqrt{\frac{1 - R^2}{n - 2}}\) | \(t(df = n - 2)\) | \(n \geq 30\) or data are normal, R \(\approx 0\) |