11.4: Chapter 11 Formulas
- Page ID
- 34779
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)One-Way ANOVA
\(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \ldots = \mu_{k}\)
f\(H_{1}:\) At least one mean is different.
| Source | \(SS\) = Sum of Squares | \(df\) | \(MS\) = Mean Square | F |
|---|---|---|---|---|
| Between (Factor) | \(\sum n_{i} \left(\bar{x}_{i} - \bar{x}_{GM}\right)^{2}\) | \(k-1\) | \(MSB = \frac{SSB}{k-1}\) | \(F = \frac{MSB}{MSW}\) |
| Within (Error) | \(\sum \left(n_{i} - 1\right) s_{i}^{2}\) | \(N-k\) | \(MSW = \frac{SSW}{N-k}\) | |
| Total | SST | \(N-1\) |
\(\bar{x}_{i}\) = sample mean from the \(i^{th}\) group
\(n_{i}\) = sample size of the \(i^{th}\) group
\(k\) = number of groups
\(s_{i}^{2}\) = sample variance from the \(i^{th}\) group
\(N = n_{1} + n_{2} + \ldots + n_{k}\)
\(\bar{x}_{GM} = \frac{\sum x_{i}}{N}\)
Bonferroni Test
\(H_{0}: \mu_{i} = \mu_{j}\)
\(H_{1}: \mu_{i} \neq \mu_{j}\)
Bonferroni test statistic: \(t = \dfrac{\bar{x}_{i} - \bar{x}_{j}}{\sqrt{ \left(MSW \left(\frac{1}{n_{i}} + \frac{1}{n_{j}}\right) \right)}}\)
Multiply p-value by \(m = {}_k C_{2}\), divide area for critical value by \(m = {}_k C_{2}\).
Two-Way ANOVA
| Row Effect (Factor A): |
\(H_{0}:\) The row variable has no effect on the average ___________________. \(H_{1}:\) The row variable has an effect on the average ___________________. |
| Column Effect (Factor B): |
\(H_{0}\): The column variable has no effect on the average ___________________. \(H_{1}\): The column variable has an effect on the average ___________________. |
| Interaction Effect (A×B): |
\(H_{0}:\) There is no interaction effect between row variable and column variable on the average ___________________. \(H_{1}:\) There is an interaction effect between row variable and column variable on the average ___________________. |
| Source | \(SS\) | \(df\) | \(MS\) | F |
|---|---|---|---|---|
| \(A\) (row factor) | \(SS_{A}\) | \(a-1\) | \(MS_{A} = \frac{SS_{A}}{df_{A}}\) | \(F_{A} = \frac{MS_{A}}{MSE}\) |
| \(B\) (column factor) | \(SS_{B}\) | \(b-1\) | \(MS_{B} = \frac{SS_{B}}{df_{B}}\) | \(F_{B} = \frac{MS_{B}}{MSE}\) |
| \(A \times B\) (interaction) | \(SS_{A \times B}\) | \((a-1)(b-1)\) | \(MS_{A \times B} = \frac{SS_{A \times B}}{df_{A \times B}}\) | \(F_{A \times B} = \frac{MS_{A \times B}}{MSE}\) |
| Error (within) | \(SSE\) | \(ab(n-1)\) | \(MSE = \frac{SSE}{df_{E}}\) | |
| Total | \(SST\) | \(N-1\) |