Introduction to Statistics, Lecture 11 - Regression Analysis (Chapter ...

The Method of Least SquaresThe Method of Least SquaresThe Method of Least SquaresThe Method of Least Squaresa and b are determined byb = S xyS xxa = ȳ − b · ¯xa and b are the values that give the regression line thatminimizes the squared distance between the points andthe linea is an estimate for α and b is an estimate for βPer Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 11 Fall 2012 17 / 32In the example we getS xx =S yy =S xy =n∑(x i − ¯x) 2 = 143i=1n∑(y i − ȳ) 2 = 31533i=1n∑(x i − ¯x)(y i − ȳ) = 2119i=1along with ¯x = 6.50 and ȳ = 100.67Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 11 Fall 2012 18 / 32The Method of Least SquaresThe Method of Least SquaresInferences for the Regression ModelInferences for the Regression ModelEstimates for α and β:The model is:b = S xyS xx= 2119143 = 14.82a = ȳ − b · ¯x = 100.67 − 14.82 · 6.50 = 4.34ŷ = 4.34 + 14.82 · xWe assume that the observed data (Y i , x i ) can bedescribed by the modelY i = α + βx i + ε iwhere it is assumed that ε i are independent normallydistributed stochastic variables with mean 0 and constantvariance σ 2An estimate of σ 2 iss 2 e = S yy − (S xy ) 2 /S xxn − 2Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 11 Fall 2012 19 / 32Per Bruun Brockhoff (pbb@imm.dtu.dk) Introduction to Statistics, Lecture 11 Fall 2012 21 / 32