ST551 Final Formula Sheet

This is a draft. You may email me (Charlotte) to suggest additions.

Pooled variance

\[ \begin{aligned} s_p^2 = \hat{\sigma}^2 = \frac{(n-1)s_Y^2 + (m-1)s_X^2}{n + m - 2} \end{aligned} \]

CI for median, based on Sign Test

\[ \begin{aligned} \Biggl( \left(\frac{n - z_{1-\alpha/2}\sqrt{n}}{2} \right)^{\text{th}} \text{ smallest observation}, \\ \left(\frac{n + z_{1-\alpha/2}\sqrt{n}}{2} + 1 \right)^{\text{th}} \text{ smallest observation}\Biggr) \end{aligned} \]

Welch-Satterthwaite degrees of freedom

\[ v = \frac{(s_Y^2/n + s_X^2/m)^2}{\frac{s_Y^4}{n^2(n-1)} + \frac{s_X^4}{m^2(m-1)} } \]

Mantel-Haenszel test

\[ X = \frac{\left(\sum_{j = 1}^{k}( a_j - E(a_j)) \right)^2}{\sum_{j = 1}^{k} Var(a_j)} \]

\[ E(a_j) = \frac{(R_{1j})(C_{1j})}{N_j} \]

\[ Var(a_j) = \frac{R_{1j}C_{1j}R_{2j}C_{2j}}{N_j^2(N_j - 1)} \]