Below is a matrix of correlations among three variables x1 x

Below is a matrix of correlations among three variables, x_1 x_2 arid y. The standard deviations of these three variables are: s_x1, = 11.88, s_x2 = 6.02, and sy = 9.95. Use these correlations and standard deviations to compute a regression formula to predict the value of y from x_1, and x_2. Show your calculations. Use the regression equation to predict the value of y when x_1, is 30 and x_2 is 6.

Solution

x=rbind(c(1.0000,0.762383, -0.829257),c(0.762383, 1.0000,-0.3596465), c(-0.829257, -0.3596465,1.000))
sx1=11.88;sx2=6.02;sy=9.95
y1=rbind(c(x[1,1]*sx1*sx1,x[1,2]*sx1*sx2, x[1,3]*sx1*sy),c(x[2,1]*sx1*sx2,x[2,2]*sx2*sx2, x[2,3]*sx2*sy),
   c(x[3,1]*sy*sx1,x[3,2]*sy*sx2, x[3,3]*sy*sy))
b1=(y1[2,2]*y1[1,3]-y1[1,2]*y1[2,3])/(y1[1,1]*y1[2,2]-(y1[1,2])^2)

-1.110134
b2=(y1[1,1]*y1[2,3]-y1[1,2]*y1[1,3])/(y1[1,1]*y1[2,2]-(y1[1,2])^2)

1.075768
x1=30; x2=6
y=b1*x1+b2*x2

-26.84942