Question

Consider the following data for a dependent variable y and two independent variables, 21 and 32.

Answer 1

Solution:

a) estimated regression equation relating y to x1

  Y= 47.6558 + 1.9087X

For x=45 find y

Y= 47.6558 + 1.9087*45

Y= 133.54

Calculation:

b) estimated regression equation relating to x2

X*Y X² Y2 2726 841 8836 108 5076 2209 11664 112 2688 12544 576 2500 179 8950 32041 3800 1600 9025 176 8976 2601 30976 171 12654 5476 29241 118 4366 1369 13924 143 20449 8580 16112 73928 3600 5776 212 5776 44944 Sum = 488 1408 26548 213644 Based on the above table, the following is calculated: x = { x: = 488 – 48.8 P = Y = 1408 = 140.8 AUC - = 140.8. 10 12 $xx = X;) = 26548 – 4882 /10 = 2733.6 = 213644 - = 15397.6 $$y = 3x -+ () = 21901 – 1909 10 – 15976 SSxy = xy - 2x)(2x) = 3928 – 48 x 1408/10 = 5217.6 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: . SSxy m=SSxx 5217.6 2733.6 a = 1.9087 n=- : m = 140.8 – 48.8 x 1.9087 = 47.6558 Therefore, we find that the regression equation is: Y = 47.6558 +1.9087X

Y= 77.7019 + 4.7088 X

For x= 15 find y

  Y= 77.7019 + 4.7088*15

Y= 148.33

calculation:

X*Y V2 1128 144 8836 1188 121 11664 2016 324 12544 3043 289 32041 570 9025 3344 361 30976 1368 29241 118 1416 144 13924 143 2002 196 20449 212 289 44944 3604 19679 Sum = 134 1408 1968 213644 Based on the above table, the following is calculated: * = { x: = 136 = 13.4 = 13.4 1408 = 140.8 10 = 1968 - = 172.4 i=1 55«x = *-- (2x) = 1905 – 134° 10 = 12.4 56 = + (*) * 21901 – 14087/10 – 15976 56&v = Şxx -: (**) (2x) = 1070 – 194 x 108/10 - S Sys = 213644 - 1 - 15397.6 i=1 SS = 19679 – 134 x 140 = 811.8 i=1 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: m = SSxy 811.8 = 4.7088 SSxx 172.4 n=Y - Ž. m = 140.8 – 13.4 x 4.7088 = 77.7019 Therefore, we find that the regression equation is: Y = 77.7019 +4.7088X
c) estimated regression equation relating y to x1 and x2

Y= −19.0534 + 1.9358X1​ + 4.8795X2

For x1=45 and x2=15 find y

  Y=−19.0534 + 1.9358*45 + 4.8795*15

Y= 141.25

Calculation:

These are the data that have been provided for the dependent and independent variables: 108 112 179 95 176 17 118 143 212 The following matrix and vector are defined in order to conduct the matrix calculation required to compute the estimated multiple regression coefficients: [i 29 127 [ 941 1 47 11 1 50 17 X = 1 51 19 · y = 1 60 14 1 76 17 Now, the vector with the estimated regression coefficients ß is computed through the following matrix operation: @ = (X'X)-{X'y

Now, for the values provided for X and y, we get that the vector with estimated regression coefficients is computed as follows: 3 = (xx)-1x/y [111 111 11 [[111 111 111 17 ། |29 47 24 50 40 51 74 37 60 76 [ |12 11 18 17 6 19 8 12 14 17 29 47 24 50 40 51 74 37 12 11 18 176 198 12 1 60 14 1 76 17 ༼ 104881347-1[1408] 488 26548 6524] . |73928 [1346524 1968 | 19679 2.055846 -0.018293 —0.079339] [ 1408 ] |-0.018293 0.0003663.2E – 5 [ 73928 -0.0793393 .2E – 5 0.005803 [19679 [-19.0534] [Bo 1.9358 4.8795 Therefore, based on the data provided, the estimated multiple linear regression equation is: _Y = -19.0534 + 1.9358X1 + 4.8795X2