Roll no: C91/STS/191016

"" "" # Importing Datasets:

data1=read.csv("C:/Users/ww/Downloads/Multiv_prac_2020.csv")
head(data)
##                                                                             
## 1 function (..., list = character(), package = NULL, lib.loc = NULL,        
## 2     verbose = getOption("verbose"), envir = .GlobalEnv, overwrite = TRUE) 
## 3 {                                                                         
## 4     fileExt <- function(x) {                                              
## 5         db <- grepl("\\\\.[^.]+\\\\.(gz|bz2|xz)$", x)                     
## 6         ans <- sub(".*\\\\.", "", x)

Renaming Variables:

X=data1[,2:7]
colnames(X)=c("X1","X2","X3","x4","X5","X6")
head(X)

(a) Test at 5% level of significanceif the mean vector \(\mu\) = (\(\mu\) 1, \(\mu\) 2, \(\mu\) 3, \(\mu\) 4, \(\mu\) 5, \(\mu\) 6)’ can be taken to be equal to (11.5, 90.0, 45.2, 4.0, 75.8, 92.4).

Calculating mean of each variable:

mean=matrix(colMeans(X),ncol=1)
mean
##          [,1]
## [1,]  10.3252
## [2,] 185.0400
## [3,]  76.3400
## [4,]   4.5464
## [5,]  67.6660
## [6,] 105.9600

Calculating sample Corrected Sum of Squares and Sum of Product Matrix

A=matrix(rep(0,36),nrow=6,ncol=6,byrow=T)##SSSP matrix

for (i in 1:50)
{
    A=A+(t(X[i,])-mean)%*%t(t(X[i,])-mean)
}
A
##           X1          X2          X3          x4          X5          X6
## X1  8562.436     7865.62   10650.752   1180.2110   1367.6308   5783.3204
## X2  7865.620 20360671.92 5778411.320 -12178.7328 182622.3680 -14447.9200
## X3 10650.752  5778411.32 1745801.220  -3032.4588  66588.0780   6673.6800
## x4  1180.211   -12178.73   -3032.459    715.7846    866.0029   -346.7372
## X5  1367.631   182622.37   66588.078    866.0029  58869.5122  35849.0320
## X6  5783.320   -14447.92    6673.680   -346.7372  35849.0320 126833.9200
μ=c(11.5,90.0,45.2,4.0,75.8,92.4)
μ1=mean-μ
μ1
##         [,1]
## [1,] -1.1748
## [2,] 95.0400
## [3,] 31.1400
## [4,]  0.5464
## [5,] -8.1340
## [6,] 13.5600

Calculating Hotelling T2

T2=50*49*t(μ1)%*%solve(A)%*%μ1 ####Hotelling's T^2
T2  
##         [,1]
## [1,] 24.4207

Calculating F statistic:

F=(T2/49)*(44/6)
F
##          [,1]
## [1,] 3.654799

Calculating F distribution value with 6 and 44 degrees of freedom:

qf(0.95,6,44)
## [1] 2.313264