#set document(title: "12.12 Chapter 12 Formulas", author: "Rachel Webb")
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== 12.12#h(0.6em)Chapter 12 Formulas


    #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "S S sub x x equals open parenthesis n minus 1 close parenthesis s sub r x squared")[$S S_(x x) = ( n − 1 ) s_(r x)^(2)$] #linebreak() 
            #math.equation(block: false, alt: "S S sub y y equals open parenthesis n minus 1 close parenthesis s sub y squared")[$S S_(y y) = ( n − 1 ) s_(y)^(2)$] #linebreak() 
            #math.equation(block: false, alt: "S S sub x y equals ∑ open parenthesis x y close parenthesis minus n times x bar times y bar")[$S S_(x y) = ∑ ( x y ) − n · overline(x) · overline(y)$]], [#strong[Correlation Coefficient] #linebreak() 
            #math.equation(block: false, alt: "r equals the fraction S S sub x y over the square root of open parenthesis S S sub x x times S S sub y y close parenthesis")[$r = frac(S S_(x y), sqrt(( S S_(x x) · S S_(y y) )))$]]),
  [#strong[Correlation t-test] #linebreak() 
            #math.equation(block: false, alt: "H sub 0 : ρ equals 0")[$H_(0) : ρ = 0$] #linebreak() 
            #math.equation(block: false, alt: "H sub 1 : ρ not equal to 0")[$H_(1) : ρ ≠ 0$] #linebreak() 
            #math.equation(block: false, alt: "t equals r the square root of open parenthesis the fraction n minus 2 over 1 minus r squared close parenthesis")[$t = r sqrt(( frac(n − 2, 1 − r^(2)) ))$] #linebreak() 
            #math.equation(block: false, alt: "d f equals n minus 2")[$d f = n − 2$]], [#strong[Regression Equation] (Line of Best Fit) #linebreak() 
            #math.equation(block: false, alt: "y hat equals b sub 0 plus b sub 1 x")[$hat(y) = b_(0) + b_(1) x$]],
  [#strong[Slope] #linebreak() 
            #math.equation(block: false, alt: "b sub 1 equals the fraction S S sub x y over S S sub x x")[$b_(1) = frac(S S_(x y), S S_(x x))$]], [#strong[y-Intercept] #linebreak() 
            #math.equation(block: false, alt: "b sub 0 equals y bar minus b sub 1 x bar")[$b_(0) = overline(y) − b_(1) overline(x)$]],
  [#strong[Slope t-test] #linebreak() 
            #math.equation(block: false, alt: "H sub 0 : β sub 1 equals 0")[$H_(0) : β_(1) = 0$] #linebreak() 
            #math.equation(block: false, alt: "H sub 1 : β sub 1 not equal to 0")[$H_(1) : β_(1) ≠ 0$] #linebreak() 
            #math.equation(block: false, alt: "t equals the fraction b sub 1 over the square root of open parenthesis the fraction M S E over S S sub x x close parenthesis")[$t = frac(b_(1), sqrt(( frac(M S E, S S_(x x)) )))$] #linebreak() 
            #math.equation(block: false, alt: "d f equals n minus p minus 1 equals n minus 2")[$d f = n − p − 1 = n − 2$]], [#strong[Slope/Model F-test] #linebreak() 
            #math.equation(block: false, alt: "H sub 0 : β sub 1 equals 0")[$H_(0) : β_(1) = 0$] #linebreak() 
            #math.equation(block: false, alt: "H sub 1 : β sub 1 not equal to 0")[$H_(1) : β_(1) ≠ 0$]],
  [#strong[Standard Error of Estimate] #linebreak() 
            #math.equation(block: false, alt: "s sub e s t equals the square root of the fraction ∑ open parenthesis y sub i minus y hat sub i close parenthesis squared over n minus 2 equals the square root of M S E")[$s_(e s t) = sqrt(frac(∑ attach(( y_(i) − hat(y)_(i) ), t: 2), n − 2)) = sqrt(M S E)$]], [#strong[Residual] #linebreak() 
            #math.equation(block: false, alt: "e sub i equals y sub i minus y hat sub i")[$e_(i) = y_(i) − hat(y)_(i)$]],
  [#strong[Prediction Interval] #linebreak() 
            #math.equation(block: false, alt: "y hat ± t sub α / 2 times s sub e s t the square root of open parenthesis 1 plus the fraction 1 over n plus the fraction open parenthesis x minus x bar close parenthesis squared over S S sub x x close parenthesis")[$hat(y) ± t_(α / 2) · s_(e s t) sqrt(( 1 + frac(1, n) + frac(attach(( x − overline(x) ), t: 2), S S_(x x)) ))$]], [#strong[Coefficient of Determination] #linebreak() 
            #math.equation(block: false, alt: "R squared equals open parenthesis r close parenthesis squared equals the fraction S S R over S S T")[$R^(2) = ( r )^(2) = frac(S S R, S S T)$]],
  [#strong[Multiple Linear Regression Equation] #linebreak() 
            #math.equation(block: false, alt: "y hat equals b sub 0 plus b sub 1 x sub 1 plus b sub 2 x sub 2 plus times plus b sub p x sub p")[$hat(y) = b_(0) + b_(1) x_(1) + b_(2) x_(2) + · + b_(p) x_(p)$]], [#strong[Model F-Test for Multiple Regression] #linebreak() 
            #math.equation(block: false, alt: "H sub 0 : β sub 1 equals β sub 2 equals ⋯ equals β sub p equals 0")[$H_(0) : β_(1) = β_(2) = ⋯ = β_(p) = 0$] #linebreak() 
            #math.equation(block: false, alt: "H sub 1 :")[$H_(1) :$] At least one slope is not zero.],
  [#strong[Adjusted Coefficient of Determination] #linebreak() 
            #math.equation(block: false, alt: "R sub a d j squared equals 1 minus open parenthesis the fraction open parenthesis 1 minus R squared close parenthesis open parenthesis n minus 1 close parenthesis over open parenthesis n minus p minus 1 close parenthesis close parenthesis")[$R_(a d j)^(2) = 1 − ( frac(( 1 − R^(2) ) ( n − 1 ), ( n − p − 1 )) )$]], [],
))


    #figure(figph[Regression ANOVA table with equations.], alt: "Regression ANOVA table with equations.", caption: none)