#set document(title: "11.4 Chapter 11 Formulas", author: "Rachel Webb")
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== 11.4#h(0.6em)Chapter 11 Formulas


    
=== One-Way ANOVA


    #math.equation(block: true, alt: "H sub 0 : μ sub 1 equals μ sub 2 equals μ sub 3 equals … equals μ sub k")[$H_(0) : μ_(1) = μ_(2) = μ_(3) = … = μ_(k)$]


    f#math.equation(block: false, alt: "H sub 1 :")[$H_(1) :$] At least one mean is different.


    #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Source], [#math.equation(block: false, alt: "S S")[$S S$] = Sum of Squares], [#math.equation(block: false, alt: "d f")[$d f$]], [#math.equation(block: false, alt: "M S")[$M S$] = Mean Square], [F]),
  [Between (Factor)], [#math.equation(block: false, alt: "∑ n sub i open parenthesis x bar sub i minus x bar sub G M close parenthesis squared")[$∑ n_(i) attach(( overline(x)_(i) − overline(x)_(G M) ), t: 2)$]], [#math.equation(block: false, alt: "k minus 1")[$k − 1$]], [#math.equation(block: false, alt: "M S B equals the fraction S S B over k minus 1")[$M S B = frac(S S B, k − 1)$]], [#math.equation(block: false, alt: "F equals the fraction M S B over M S W")[$F = frac(M S B, M S W)$]],
  [Within (Error)], [#math.equation(block: false, alt: "∑ open parenthesis n sub i minus 1 close parenthesis s sub i squared")[$∑ ( n_(i) − 1 ) s_(i)^(2)$]], [#math.equation(block: false, alt: "N minus k")[$N − k$]], [#math.equation(block: false, alt: "M S W equals the fraction S S W over N minus k")[$M S W = frac(S S W, N − k)$]], [],
  [#strong[Total]], [#strong[SST]], [#strong[#math.equation(block: false, alt: "N minus 1")[$N − 1$]]], [], [],
))


    #math.equation(block: false, alt: "x bar sub i")[$overline(x)_(i)$] = sample mean from the #math.equation(block: false, alt: "i to the power t h")[$i^(t h)$] group


    #math.equation(block: false, alt: "n sub i")[$n_(i)$] = sample size of the #math.equation(block: false, alt: "i to the power t h")[$i^(t h)$] group


    #math.equation(block: false, alt: "k")[$k$] = number of groups


    #math.equation(block: false, alt: "s sub i squared")[$s_(i)^(2)$] = sample variance from the #math.equation(block: false, alt: "i to the power t h")[$i^(t h)$] group


    #math.equation(block: true, alt: "N equals n sub 1 plus n sub 2 plus … plus n sub k")[$N = n_(1) + n_(2) + … + n_(k)$]


    #math.equation(block: true, alt: "x bar sub G M equals the fraction ∑ x sub i over N")[$overline(x)_(G M) = frac(∑ x_(i), N)$]


    
=== Bonferroni Test


    #math.equation(block: true, alt: "H sub 0 : μ sub i equals μ sub j")[$H_(0) : μ_(i) = μ_(j)$]


    #math.equation(block: true, alt: "H sub 1 : μ sub i not equal to μ sub j")[$H_(1) : μ_(i) ≠ μ_(j)$]


    Bonferroni test statistic: #math.equation(block: false, alt: "t equals the fraction x bar sub i minus x bar sub j over the square root of open parenthesis M S W open parenthesis the fraction 1 over n sub i plus the fraction 1 over n sub j close parenthesis close parenthesis")[$t = frac(overline(x)_(i) − overline(x)_(j), sqrt(( M S W ( frac(1, n_(i)) + frac(1, n_(j)) ) )))$]


    Multiply p-value by #math.equation(block: false, alt: "m equals k C sub 2")[$m = k C_(2)$], divide area for critical value by #math.equation(block: false, alt: "m equals k C sub 2")[$m = k C_(2)$].


    
=== Two-Way ANOVA


    #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Row Effect (Factor A): #linebreak()], [#math.equation(block: false, alt: "H sub 0 :")[$H_(0) :$] The row variable has no effect on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. #linebreak() 
            #math.equation(block: false, alt: "H sub 1 :")[$H_(1) :$] The row variable has an effect on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.]),
  [Column Effect (Factor B): #linebreak()], [#math.equation(block: false, alt: "H sub 0")[$H_(0)$]: The column variable has no effect on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. #linebreak() 
            #math.equation(block: false, alt: "H sub 1")[$H_(1)$]: The column variable has an effect on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.],
  [Interaction Effect (A×B): #linebreak() 
             #linebreak() 
             #linebreak()], [#math.equation(block: false, alt: "H sub 0 :")[$H_(0) :$] There is no interaction effect between row variable and column variable on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. #linebreak() 
            #math.equation(block: false, alt: "H sub 1 :")[$H_(1) :$] There is an interaction effect between row variable and column variable on the average \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.],
))


    #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Source], [#math.equation(block: false, alt: "S S")[$S S$]], [#math.equation(block: false, alt: "d f")[$d f$]], [#math.equation(block: false, alt: "M S")[$M S$]], [F]),
  [#math.equation(block: false, alt: "A")[$A$] (row factor)], [#math.equation(block: false, alt: "S S sub A")[$S S_(A)$]], [#math.equation(block: false, alt: "a minus 1")[$a − 1$]], [#math.equation(block: false, alt: "M S sub A equals the fraction S S sub A over d f sub A")[$M S_(A) = frac(S S_(A), d f_(A))$]], [#math.equation(block: false, alt: "F sub A equals the fraction M S sub A over M S E")[$F_(A) = frac(M S_(A), M S E)$]],
  [#math.equation(block: false, alt: "B")[$B$] (column factor)], [#math.equation(block: false, alt: "S S sub B")[$S S_(B)$]], [#math.equation(block: false, alt: "b minus 1")[$b − 1$]], [#math.equation(block: false, alt: "M S sub B equals the fraction S S sub B over d f sub B")[$M S_(B) = frac(S S_(B), d f_(B))$]], [#math.equation(block: false, alt: "F sub B equals the fraction M S sub B over M S E")[$F_(B) = frac(M S_(B), M S E)$]],
  [#math.equation(block: false, alt: "A times B")[$A × B$] (interaction)], [#math.equation(block: false, alt: "S S sub A times B")[$S S_(A × B)$]], [#math.equation(block: false, alt: "open parenthesis a minus 1 close parenthesis open parenthesis b minus 1 close parenthesis")[$( a − 1 ) ( b − 1 )$]], [#math.equation(block: false, alt: "M S sub A times B equals the fraction S S sub A times B over d f sub A times B")[$M S_(A × B) = frac(S S_(A × B), d f_(A × B))$]], [#math.equation(block: false, alt: "F sub A times B equals the fraction M S sub A times B over M S E")[$F_(A × B) = frac(M S_(A × B), M S E)$]],
  [Error (within)], [#math.equation(block: false, alt: "S S E")[$S S E$]], [#math.equation(block: false, alt: "a b open parenthesis n minus 1 close parenthesis")[$a b ( n − 1 )$]], [#math.equation(block: false, alt: "M S E equals the fraction S S E over d f sub E")[$M S E = frac(S S E, d f_(E))$]], [],
  [Total], [#math.equation(block: false, alt: "S S T")[$S S T$]], [#math.equation(block: false, alt: "N minus 1")[$N − 1$]], [], [],
))