Data

Data yang digunakan dalam praktikum ini adalah data Perusahaan toluca.

Perusahaan toluca ingin memprediksi berapa lama waktu yang dibutuhkan dalam bekerja berdasarkan banyaknya jumlah produksi yang dikerjakan.

library(readxl)
toluca<-read_excel("D:/toluca.xlsx")
head(toluca)

## # A tibble: 6 × 2
##   LotSize WorkHours
##     <dbl>     <dbl>
## 1      80       399
## 2      30       121
## 3      50       221
## 4      90       376
## 5      70       361
## 6      60       224

Eksplorasi Data

summary(toluca)

##     LotSize      WorkHours    
##  Min.   : 20   Min.   :113.0  
##  1st Qu.: 50   1st Qu.:224.0  
##  Median : 70   Median :342.0  
##  Mean   : 70   Mean   :312.3  
##  3rd Qu.: 90   3rd Qu.:389.0  
##  Max.   :120   Max.   :546.0

plot(toluca$LotSize, toluca$WorkHours,xlab="LotSize",ylab="WorkHours",pch=16)

Persamaan Regresi

model2 <- lm(WorkHours~LotSize, data=toluca)
summary(model2)

## 
## Call:
## lm(formula = WorkHours ~ LotSize, data = toluca)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -83.876 -34.088  -5.982  38.826 103.528 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   62.366     26.177   2.382   0.0259 *  
## LotSize        3.570      0.347  10.290 4.45e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 48.82 on 23 degrees of freedom
## Multiple R-squared:  0.8215, Adjusted R-squared:  0.8138 
## F-statistic: 105.9 on 1 and 23 DF,  p-value: 4.449e-10

Persaman Regresi dapat dituliskan sebagai berikut:

\(\hat{y}=b_{0}+b_{1}X\)

\(\hat{y}=62.366+3.570X\)

ANOVA

anova(model2)

## Analysis of Variance Table
## 
## Response: WorkHours
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## LotSize    1 252378  252378  105.88 4.449e-10 ***
## Residuals 23  54825    2384                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interval Kepercayaan

confint (model2, level = 0.95)

##                2.5 %     97.5 %
## (Intercept) 8.213711 116.518006
## LotSize     2.852435   4.287969

Persamaan Regresi dengan Matriks

Y<-matrix(toluca$WorkHours)
N<-nrow(Y) #mengetahui jumlah baris Y
X<-matrix(1,N)
X<-cbind(X,toluca$LotSize)
J<-matrix(1,N,N)
Xt<-t(X)
A<-Xt %*% X
A

##      [,1]   [,2]
## [1,]   25   1750
## [2,] 1750 142300

B<-solve(A)
B

##              [,1]          [,2]
## [1,]  0.287474747 -3.535354e-03
## [2,] -0.003535354  5.050505e-05

C<- Xt %*% Y
C

##        [,1]
## [1,]   7807
## [2,] 617180

Beta<-B %*% C
Beta

##           [,1]
## [1,] 62.365859
## [2,]  3.570202

Dari perhitungan di atas jelas bahwa nilai b0 = 62.365859 dan nilai b1 = 3.570202.

Inferensi

JKT<-(t(Y)%*%Y)-((1/N)*(t(Y)%*%J%*%Y))
JKT

##        [,1]
## [1,] 307203

JKG<-(t(Y)%*%Y)-(t(Beta)%*%t(X)%*%Y)
JKG

##          [,1]
## [1,] 54825.46

JKR<-JKT-JKG
JKR

##          [,1]
## [1,] 252377.6

fhitung<-(JKR/(2-1))/(JKG/(N-2))
fhitung

##          [,1]
## [1,] 105.8757

qf(1-0.01,(2-1),(N-2)) #ftabel alpha = 0.01

## [1] 7.881134

KTG<-JKG/(N-2)
KTG<-as.numeric(KTG)
KTG

## [1] 2383.716

Sb<-KTG*solve(t(X)%*%X)
Sb

##            [,1]       [,2]
## [1,] 685.258045 -8.4272774
## [2,]  -8.427277  0.1203897

thitung0<-Beta[1,1]/sqrt(Sb[1,1])
thitung0

## [1] 2.382428

thitung1<-Beta[2,1]/sqrt(Sb[2,2])
thitung1

## [1] 10.28959

qt(1-0.01/2,(N-2)) #ttabel alpha = 0.01

## [1] 2.807336

Data

library(readxl)
bodyf<- read_excel("D:/data_bodyfat.xlsx")
print.data.frame(bodyf[,-1]) #menghapus kolom 1

##    Triceps_X1 Thigh_X2 Midarm_X3 Bodyfat_Y
## 1        19.5     43.1      29.1      11.9
## 2        24.7     49.8      28.2      22.8
## 3        30.7     51.9      37.0      18.7
## 4        29.8     54.3      31.1      20.1
## 5        19.1     42.2      30.9      12.9
## 6        25.6     53.9      23.7      21.7
## 7        31.4     58.5      27.6      27.1
## 8        27.9     52.1      30.6      25.4
## 9        22.1     49.9      23.2      21.3
## 10       25.5     53.5      24.8      19.3
## 11       31.1     56.6      30.0      25.4
## 12       30.4     56.7      28.3      27.2
## 13       18.7     46.5      23.0      11.7
## 14       19.7     44.2      28.6      17.8
## 15       14.6     42.7      21.3      12.8
## 16       29.5     54.4      30.1      23.9
## 17       27.7     55.3      25.7      22.6
## 18       30.2     58.6      24.6      25.4
## 19       22.7     48.2      27.1      14.8
## 20       25.2     51.0      27.5      21.1

dengan,

X1: Triceps Skinfold Thickness

X2: Thigh Circumference

X3: Midarm Circumference

Y: Body Fat

Persamaan Regresi

modela<- lm(formula = Bodyfat_Y ~ Triceps_X1+Thigh_X2+Midarm_X3, data=bodyf)
summary(modela)

## 
## Call:
## lm(formula = Bodyfat_Y ~ Triceps_X1 + Thigh_X2 + Midarm_X3, data = bodyf)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7263 -1.6111  0.3923  1.4656  4.1277 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  117.085     99.782   1.173    0.258
## Triceps_X1     4.334      3.016   1.437    0.170
## Thigh_X2      -2.857      2.582  -1.106    0.285
## Midarm_X3     -2.186      1.595  -1.370    0.190
## 
## Residual standard error: 2.48 on 16 degrees of freedom
## Multiple R-squared:  0.8014, Adjusted R-squared:  0.7641 
## F-statistic: 21.52 on 3 and 16 DF,  p-value: 7.343e-06

ANOVA

anova (modela)

## Analysis of Variance Table
## 
## Response: Bodyfat_Y
##            Df Sum Sq Mean Sq F value    Pr(>F)    
## Triceps_X1  1 352.27  352.27 57.2768 1.131e-06 ***
## Thigh_X2    1  33.17   33.17  5.3931   0.03373 *  
## Midarm_X3   1  11.55   11.55  1.8773   0.18956    
## Residuals  16  98.40    6.15                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interval Kepercayaan

confint (modela, level = 0.95)

##                  2.5 %     97.5 %
## (Intercept) -94.444550 328.613940
## Triceps_X1   -2.058507  10.726691
## Thigh_X2     -8.330476   2.616780
## Midarm_X3    -5.568367   1.196247

Persamaan Regresi dengan Matriks

Y<-matrix(bodyf$Bodyfat_Y)
N<-nrow(Y) #mengetahui jumlah baris Y
X<-matrix(1,N)
X<-cbind(X,bodyf$Triceps_X1,bodyf$Thigh_X2, bodyf$Midarm_X3)
J<-matrix(1,N,N)
Xt<-t(X)
A<-Xt %*% X
A

##        [,1]     [,2]     [,3]     [,4]
## [1,]   20.0   506.10  1023.40   552.40
## [2,]  506.1 13286.29 26358.69 14137.83
## [3,] 1023.4 26358.69 52888.00 28297.02
## [4,]  552.4 14137.83 28297.02 15510.02

B<-solve(A)
B

##            [,1]       [,2]        [,3]        [,4]
## [1,] 1618.86721 48.8102522 -41.8487041 -25.7987855
## [2,]   48.81025  1.4785133  -1.2648388  -0.7785022
## [3,]  -41.84870 -1.2648388   1.0839791   0.6657581
## [4,]  -25.79879 -0.7785022   0.6657581   0.4139009

C<- Xt %*% Y
C

##          [,1]
## [1,]   403.90
## [2,] 10631.65
## [3,] 21113.50
## [4,] 11206.12

Beta<-B %*% C
Beta

##            [,1]
## [1,] 117.084695
## [2,]   4.334092
## [3,]  -2.856848
## [4,]  -2.186060

Inferensi

JKT<-(t(Y)%*%Y)-((1/N)*(t(Y)%*%J%*%Y))
JKT

##          [,1]
## [1,] 495.3895

JKG<-(t(Y)%*%Y)-(t(Beta)%*%t(X)%*%Y)
JKG

##          [,1]
## [1,] 98.40489

JKR<-JKT-JKG
JKR

##          [,1]
## [1,] 396.9846

fhitung<-(JKR/(4-1))/(JKG/(N-4))
fhitung

##          [,1]
## [1,] 21.51571

qf(0.99,(4-1),(N-4)) #alfa=1%

## [1] 5.292214

KTG<-JKG/(N-4)
KTG<-as.numeric(KTG)
KTG

## [1] 6.150306

Sb<-KTG*solve(t(X)%*%X)
Sb

##           [,1]       [,2]        [,3]        [,4]
## [1,] 9956.5279 300.197962 -257.382315 -158.670413
## [2,]  300.1980   9.093309   -7.779145   -4.788026
## [3,] -257.3823  -7.779145    6.666803    4.094615
## [4,] -158.6704  -4.788026    4.094615    2.545617

thitung0<-Beta[1,1]/sqrt(Sb[1,1])
thitung0

## [1] 1.1734

thitung1<-Beta[2,1]/sqrt(Sb[2,2])
thitung1

## [1] 1.437266