can graph time The cost of houses sold over an average income of people as a function of their years of formaleducation incidence of strokes as a function of blood pressure heremare so many points to plot how can we understand this data amount of errors in data assessments poooredictions Wenhitallpoints point on the line y AxtB w x coordinate vertical distance between the y coordinate AxitB and Yi To avoid confusing distance E direction we can square of vertical distance or the opposite E E Ax B Y D E E E on 2 5 3 8 error EN x y is large a large error E A B 2AtB 3AtB respectively Ciel if all points are E E least squares in approximating the data points 1 4 the straight line by y mxtb 1 2 3 the y coordinate of the corresponding point of the line is Thetotal error An IN by the line y AxtB is usually measured by the sum E of the squares or regression line the line E o if a point is far away from the line E extFindthe straightline that minimizes the least squares error for the points hen The squares of the vertical distances from the points E Then E FA 2 ATB 4 212A B 5 2 213A B 8 3 2A 2B 8 8A 4B 20 18A GB 48 28A 12B 3 2 ATB 4 1 a Error depends on the choice of A B flA B wantto find values At B that minimize f A B 2A 2B 8 4A 2B 10 6A 2B 16 12 A ii iii take partial derivatives W respectto At B 212A B 5 4 213A B 8 1 fEE E A B 4 least squares err 3A B 812 2A B 512 1212 6B 34 3 8 Are 6 B 34 0 EE 6851536 3AtB 87 2At B 57 A B 4 1 4 2 5 Ez 76 E E E 0 ex The following table gives the number accident X yitxzy to NYN it XN related deaths in the U s for certain years in thousands of car obtain fits a Use the formula above to the straight line that best this data b use the line from A to estimate the deaths in 2012 number of car accident related 0 10 15 17 18 19 Ex 79 46 8 43 4 45 3 43 9 39 7 35 9 9 255 0 434 679 5 0 100 225 289 324 361 746 3 714 6 682 1 2 9 3256 5 x 1299 B FI 255t9AI 47 64 6l I25I 0 39 the equation of the least squares is i 0 39 47 64 y 0 397 22 47 64 I 39 06 estimated deaths are related accidental car of 39 060 in 2012 A A Y ex x 0 y 515 5 5 15 10 7 25 16 7 25 y 0 25 75 72 5 116 x 0 25 100 256 2 31 29 24 8 2 9 214 25 2 2 381 Ex 961 3 y 8 let 0 157 8 4 983 6 239 6 24148112006 11 4 t NEE EE B Ey II I 19 9 8 3 0 157 1 4 0 157 4 9836 4 98325 4 983 c 10 0 157 4 983 5 017 6 1574 31 9554 32 Years 20326 subject to the constraint 8 ftp o g glx y 2 392 10 18 9 4 2 3 2x 69 4 69 39 mi F x y X I Fix g 2x Ex F IF 8 x y by 0 8 3y 9 8 49 216 7120 2 342 10 62 3127 10 36 12 10 58J minimization