data <- read.csv("C:\\Users\\ASUS\\Documents\\Nita\\SEMESTER 4\\Analisis Regresi\\Kuliah\\Data Anreg Berganda 2.csv", sep=";")
data
##      Y  X1  X2  X3 X4 X5  X6
## 1  443  49  79  76  8 15 205
## 2  290  27  70  31  6  6 129
## 3  676 115  92 130  0  9 339
## 4  536  92  62  92  5  8 247
## 5  481  67  42  94 16  3 202
## 6  296  31  54  34 14 11 119
## 7  453 105  60  47  5 10 212
## 8  617 114  85  84 17 20 285
## 9  514  98  72  71 12 -1 242
## 10 400  15  59  99 15 11 174
## 11 473  62  62  81  9  1 207
## 12 157  25  11   7  9  9  45
## 13 440  45  65  84 19 13 195
## 14 480  92  75  63  9 20 232
## 15 316  27  26  82  4 17 134
## 16 530 111  52  93 11 13 256
## 17 610  78 102  84  5  7 266
## 18 617 106  87  82 18  7 276
## 19 600  97  98  71 12  8 266
## 20 480  67  65  62 13 12 196
## 21 279  38  26  44 10  8 110
## 22 446  56  32  99 16  8 188
## 23 450  54 100  50 11 15 205
## 24 335  53  55  60  8  0 170
## 25 459  61  53  79  6  5 193
## 26 630  60 108 104 17  8 273
## 27 483  83  78  71 11  8 233
## 28 617  74 125  66 16  4 265
## 29 605  89 121  71  8  8 283
## 30 388  64  30  81 10 10 176
## 31 351  34  44  65  7  9 143
## 32 366  71  34  56  8  9 162
## 33 493  88  30  87 13  0 207
## 34 648 112 105 123  5 12 340
## 35 449  57  69  72  5  4 200
## 36 340  61  35  55 13  0 152
## 37 292  29  45  47 13 13 123
## 38 688  82 105  81 20  9 268
## 39 408  80  55  61 11  1 197
## 40 461  82  88  54 14  7 225
n <- 40
p <- 4

n = 40 merupakan banyaknya data yang digunakan p = 4 (beta 0, beta 1, beta 2, beta 3)

Scatter Plot

plot(x = data$X1,y = data$Y)

plot(x = data$X2,y = data$Y)

plot(x = data$X3,y = data$Y)

plot(x = data$X4,y = data$Y)

plot(x = data$X5,y = data$Y)

plot(x = data$X6,y = data$Y)

(model <- lm(Y ~ X1+X2+X3, data=data))
## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = data)
## 
## Coefficients:
## (Intercept)           X1           X2           X3  
##      61.925        1.637        2.177        2.017
summary(model)
## 
## Call:
## lm(formula = Y ~ X1 + X2 + X3, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -73.919 -15.681  -4.493  22.570  99.903 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  61.9253    18.1589   3.410  0.00162 ** 
## X1            1.6365     0.2208   7.413 9.50e-09 ***
## X2            2.1769     0.2028  10.734 9.05e-13 ***
## X3            2.0173     0.2398   8.411 5.10e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 31.63 on 36 degrees of freedom
## Multiple R-squared:  0.9408, Adjusted R-squared:  0.9359 
## F-statistic: 190.7 on 3 and 36 DF,  p-value: < 2.2e-16

Mencari Nilai U

(model_Y_X1 <- lm(Y~X1,data=data))
## 
## Call:
## lm(formula = Y ~ X1, data = data)
## 
## Coefficients:
## (Intercept)           X1  
##     227.552        3.451
(u <- data$Y - model_Y_X1$coefficients[1] - (model_Y_X1$coefficients[2]*data$X1))
##  [1]   46.327313  -30.740910   51.531982   -9.084796   22.201314  -38.546687
##  [7] -136.953574   -4.016573  -51.793463  120.676423   31.458536 -156.838021
## [13]   57.133091  -65.084796   -4.740910  -80.662240  113.235425   23.594982
## [19]   37.657982   21.201314  -79.706798   25.167202   36.070091  -75.478464
## [25]   20.909980  195.361425  -31.021797  134.041203   70.269537  -60.444353
## [31]    6.098979 -106.604464  -38.279019   33.886315   24.715758  -98.090020
## [37]  -35.643799  177.429648  -95.667464  -49.570352

Mencari Nilai V

(model_X3_X1 <- lm(X3~X1, data=data))
## 
## Call:
## lm(formula = X3 ~ X1, data = data)
## 
## Coefficients:
## (Intercept)           X1  
##     45.6864       0.3873
(v <- data$X3 - model_X3_X1$coefficients[1] - (model_X3_X1$coefficients[2]*data$X1))
##  [1]  11.334449 -25.144292  39.770674  10.679263  22.362511 -23.693612
##  [7] -39.356026  -5.841996 -12.644717  47.503667  11.299160 -48.369633
## [13]  20.883769 -18.320737  25.855708   4.319994   8.101881  -4.743356
## [19] -12.257387  -9.637489 -16.404922  31.623140 -16.602200  -6.214870
## [25]   9.686490  35.073820  -6.834768  -8.348799  -9.158748  10.524500
## [31]   6.144398 -17.186809   7.228582  33.932664   4.235810 -14.313510
## [37]  -9.918952   3.552562 -15.672778 -23.447438

Menjawab Soal

4.14 bagian A

(beta3 <- sum(u*v)/sum(v^2))
## [1] 2.332771

4.14 bagian B

e_kuadrat <- sum(model$residuals^2)
sigma_kuadrat <- e_kuadrat/(n-p)
sigma <- sqrt(sigma_kuadrat)

(se_beta3 <- sigma / sqrt(sum(v^2)))
## [1] 0.2380304

Keterangan

Melalui rumus manual, didapatkan nilai beta 3 adalah 2.3327 dan standard error adalah 0.23803 Sedangkan melalui fungsi lm didapatkan nilai beta 3 adalah 2.0173 dan standard error adalah 0.2398. Adanya perbedaan tersebut dapat disebabkan oleh adanya nilai pembulatan dalam pengerjaan rumus manual dan melalui fungsi lm.