Question

Please answer this question in both mathematically and with R. I need R code also.. Its urgent...
3.14 For the variables in Table 3.4, define z 3y1 y2 2y3 (3,-1,2)y. Find z and sz in two ways: (a) Evaluate z for each row of Table 3.4 and find Z and s directly from 21,22,., 210 using (3.1) and (3.5). (b) Use za'y and s2- a'Sa, as in (3.54) and (3.55).

Answer 1

(a) R code:

y1=c(35,35,40,10,6,20,35,35,35,30)
y2=c(3.5,4.9,30,2.8,2.7,2.8,4.6,10.9,8,1.6)
y3=c(2.80,2.70,4.38,3.21,2.73,2.81,2.88,2.90,3.28,3.20)
z=3*y1-y2+2*y3
mean(z)
sum(z^2)
s=(sum(z^2)-10*mean(z)^2)/9
s

Output:

> mean(z)
[1] 83.298
> sum(z^2)
[1] 78823.5
> s=(sum(z^2)-10*mean(z)^2)/9
> s
[1] 1048.659
$\bar{z}=\frac{1}{10}\sum_{i=1}^{10}z_i= 83.298\\ \sum_{i=1}^{10}z_i^2=78823.5\\ s_z^2=\frac{1}{9}\left(\sum_{i=1}^{10}z_i^2-10\bar{z}^2 \right )=1048.659$

(b)

R code:

y1=c(35,35,40,10,6,20,35,35,35,30)
y2=c(3.5,4.9,30,2.8,2.7,2.8,4.6,10.9,8,1.6)
y3=c(2.80,2.70,4.38,3.21,2.73,2.81,2.88,2.90,3.28,3.20)
s11=(sum(y1^2)-10*mean(y1)^2)/9
s22=(sum(y2^2)-10*mean(y2)^2)/9
s33=(sum(y3^2)-10*mean(y3)^2)/9
s12=(sum(y1*y2)-10*mean(y1)*mean(y2))/9
s13=(sum(y1*y3)-10*mean(y1)*mean(y3))/9
s23=(sum(y2*y3)-10*mean(y2)*mean(y3))/9
M=matrix(c(s11,s12,s13,s12,s22,s23,s13,s23,s33),nrow=3,ncol=3)
Mean_z=c(3,-1,2)%*%c(mean(y1),mean(y2),mean(y3))
var_z=c(3,-1,2)%*%M%*%c(3,-1,2)
Mean_z
var_z

Output:

> Mean_z
[,1]
[1,] 83.298
> var_z
[,1]
[1,] 1048.659