1 Question 1:

Write the model equation of clathrate formation regressed on both surfactant and time.

--> Here Y=responser
X=predictor
X1=surfactant
X2=time $$Y=\beta_{0}+\beta_{1} X_1 + \beta_{2} X_2 + \epsilon $$

2 Question 2:

Fit the model in R without using the LM function. What is the dimensionality of Y? What is the dimensionality of X?

library(MASS)
data<-read.csv("C:\\Users\\18067\\Documents\\Fareeha Imam\\TTU R11767331\\Spring 2023\\SDA\\data-table-B8(2).csv")
X1<-data$x1
X2<-data$x2
ones<-rep(1,36)
X<-cbind(ones,X1,X2)
X
##       ones   X1  X2
##  [1,]    1 0.00  10
##  [2,]    1 0.00  50
##  [3,]    1 0.00  85
##  [4,]    1 0.00 110
##  [5,]    1 0.00 140
##  [6,]    1 0.00 170
##  [7,]    1 0.00 200
##  [8,]    1 0.00 230
##  [9,]    1 0.00 260
## [10,]    1 0.00 290
## [11,]    1 0.00  10
## [12,]    1 0.00  30
## [13,]    1 0.00  62
## [14,]    1 0.00  90
## [15,]    1 0.00 150
## [16,]    1 0.00 210
## [17,]    1 0.00 270
## [18,]    1 0.02  10
## [19,]    1 0.02  30
## [20,]    1 0.02  60
## [21,]    1 0.02  90
## [22,]    1 0.02 120
## [23,]    1 0.02 210
## [24,]    1 0.02  30
## [25,]    1 0.02  60
## [26,]    1 0.02 120
## [27,]    1 0.02 150
## [28,]    1 0.05  20
## [29,]    1 0.05  40
## [30,]    1 0.05 130
## [31,]    1 0.05 190
## [32,]    1 0.05 250
## [33,]    1 0.05  60
## [34,]    1 0.05  90
## [35,]    1 0.05 120
## [36,]    1 0.05 150
y<-data$y
d<- t(y)
Y<-t(d)
Beta<-ginv(t(X)%*%X)%*%t(X)%*%y
t(X)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## ones    1    1    1    1    1    1    1    1    1     1     1     1     1     1
## X1      0    0    0    0    0    0    0    0    0     0     0     0     0     0
## X2     10   50   85  110  140  170  200  230  260   290    10    30    62    90
##      [,15] [,16] [,17] [,18] [,19] [,20] [,21]  [,22]  [,23] [,24] [,25]  [,26]
## ones     1     1     1  1.00  1.00  1.00  1.00   1.00   1.00  1.00  1.00   1.00
## X1       0     0     0  0.02  0.02  0.02  0.02   0.02   0.02  0.02  0.02   0.02
## X2     150   210   270 10.00 30.00 60.00 90.00 120.00 210.00 30.00 60.00 120.00
##       [,27] [,28] [,29]  [,30]  [,31]  [,32] [,33] [,34]  [,35]  [,36]
## ones   1.00  1.00  1.00   1.00   1.00   1.00  1.00  1.00   1.00   1.00
## X1     0.02  0.05  0.05   0.05   0.05   0.05  0.05  0.05   0.05   0.05
## X2   150.00 20.00 40.00 130.00 190.00 250.00 60.00 90.00 120.00 150.00
dim(Y)
## [1] 36  1
dim(X)
## [1] 36  3
--> Here we can see,
Y Dimension: 36 * 1
X Dimension: 36 * 3

3 Question 3:

What are the least squares estimates of the regression parameters in the model?

Beta<-ginv(t(X)%*%X)%*%t(X)%*%y
Beta
##             [,1]
## [1,]  11.0869804
## [2,] 350.1192457
## [3,]   0.1089344
--> Here we have 3x3 Matrix and the Least Square Estimates are given below
$$Intercept=\beta_{0}$$
$$\beta_{0} = 11.0869804$$
$$Coefficients= \beta_{1} and \beta_{2}$$
$$\beta_{1}=350.1192457$$
$$\beta_{2}=0.1089344$$