Question 1:

a. Load up data mtcars and generate its descriptive statistics as shown below. Here are some packages you need to install: psych, tidyverse, tidymodels, vip, ISLR2.

library(pacman)
p_load(dplyr,psych, tidyverse, tidymodels, vip, ISLR2, kableExtra)
data("mtcars")
attach(mtcars)
mtcars %>%  kbl(caption= "Motor Trend Car Road Tests")%>% 
             row_spec(row =0, bold= TRUE, color = "black", background = "#F9EBEA") %>%
             kable_styling(bootstrap_options = "striped", full_width = F, position = "center") %>%
             kable_paper() %>%
             scroll_box(width = "100%", height = "400px")
Motor Trend Car Road Tests
mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2

b. As we need to assess the relationship between mpg and other predictors, the visualization check will be implemented. Based on the plot, which predictors may have transformations like 𝑙𝑜𝑔(𝑥), √𝑥or 𝑥^2.

pairs(mtcars)

par(mfrow = c(3, 4))

for (i in 2:ncol(mtcars)) {
    par(mar = c(3, 3, 2, 2)) # reduce the margin size
  plot(mtcars$mpg, mtcars[,i], main = paste("mpg vs.", names(mtcars)[i]))
}

From the graph below, it is able to define that the the relationship between mpg and 3 variables: disp, hp and wt which stand for displacement, Gross horsepower and Weight have transformations of 1/x. With qsec, the chart demonstrates a shape of √x while drat seems to have a linear regression relationship with mpg.

c. Run the multiple regression on mpg across all predictors and show the estimated results.

# run multiple regression of mpg on all predictors
model <- lm(mpg ~ ., data = mtcars)

# display estimated regression coefficients
summary(model)
## 
## Call:
## lm(formula = mpg ~ ., data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4506 -1.6044 -0.1196  1.2193  4.6271 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 12.30337   18.71788   0.657   0.5181  
## cyl         -0.11144    1.04502  -0.107   0.9161  
## disp         0.01334    0.01786   0.747   0.4635  
## hp          -0.02148    0.02177  -0.987   0.3350  
## drat         0.78711    1.63537   0.481   0.6353  
## wt          -3.71530    1.89441  -1.961   0.0633 .
## qsec         0.82104    0.73084   1.123   0.2739  
## vs           0.31776    2.10451   0.151   0.8814  
## am           2.52023    2.05665   1.225   0.2340  
## gear         0.65541    1.49326   0.439   0.6652  
## carb        -0.19942    0.82875  -0.241   0.8122  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared:  0.869,  Adjusted R-squared:  0.8066 
## F-statistic: 13.93 on 10 and 21 DF,  p-value: 3.793e-07

d. Using car::vif(variance inflation factor) to estimate if there is any multicollinearity among predictors. Find predictors with vif higher than 10.

Notes: Multicollinearity occurs when two or more independent variables in a data frame have a high correlation with one another in a regression model

# calculate VIFs for all predictors in the model
vifs <- car::vif(model)

# print VIFs and identify predictors with VIF > 10
print(vifs)
##       cyl      disp        hp      drat        wt      qsec        vs        am 
## 15.373833 21.620241  9.832037  3.374620 15.164887  7.527958  4.965873  4.648487 
##      gear      carb 
##  5.357452  7.908747
high_vif <- names(vifs)[vifs > 10]
if (length(high_vif) > 0) {
  message("Predictors with VIF > 10:", paste(high_vif, collapse = ", "))
} else {
  message("No predictors with VIF > 10")
}
## Predictors with VIF > 10:cyl, disp, wt

Using VIF, we can detect the strength of the correlation between the independent variables. By computing, the 3 variables cyl, disp and wt have greater value of VIF, which means they are highly affected by the other variables. In the regression model, a change of other variable might lead to the change of cyl, disp, and wt. Or by other words, they consider to be not independent variables.

e. Rerun the multiple regressions by (1) excluding disp, and (2) excluding disp and cylfrom predictors. Are there any improvements observed from the regression results?

lm1 <- lm(mpg ~ . -disp, data = mtcars)
summary(lm1)
## 
## Call:
## lm(formula = mpg ~ . - disp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.7863 -1.4055 -0.2635  1.2029  4.4753 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 12.55052   18.52585   0.677   0.5052  
## cyl          0.09627    0.99715   0.097   0.9240  
## hp          -0.01295    0.01834  -0.706   0.4876  
## drat         0.92864    1.60794   0.578   0.5694  
## wt          -2.62694    1.19800  -2.193   0.0392 *
## qsec         0.66523    0.69335   0.959   0.3478  
## vs           0.16035    2.07277   0.077   0.9390  
## am           2.47882    2.03513   1.218   0.2361  
## gear         0.74300    1.47360   0.504   0.6191  
## carb        -0.61686    0.60566  -1.018   0.3195  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.623 on 22 degrees of freedom
## Multiple R-squared:  0.8655, Adjusted R-squared:  0.8105 
## F-statistic: 15.73 on 9 and 22 DF,  p-value: 1.183e-07
lm2 <- lm(mpg ~ . -disp -cyl, data = mtcars)

The degree of freedom value for the residual term (df Residuals) is a measure of the number of observations available to estimate the error variance. The higher the value of df Residuals, the better the precision of the estimate of the error variance. We compare the value of degree of freedom and RSE between the model of all variables and the 2 new model:

summary(model)$sigma
## [1] 2.650197
summary(lm1)$sigma
## [1] 2.623418
summary(lm2)$sigma
## [1] 2.566297
summary(model)$df
## [1] 11 21 11
summary(lm1)$df
## [1] 10 22 10
summary(lm2)$df
## [1]  9 23  9

After excluding the 2 variables disp and cyl, the regression model has lower value of RSE and higher value of df, indicating that the model has better fit to the data, as it indicates that the residuals are smaller and therefore the predicted values are closer to the actual values.

Q2. This question should be answered using the data set Carseats which is available after you load uplibrary(ISLR2).

library(ISLR2)
data("Carseats")
dat <- Carseats
dat %>%  kbl(caption= "Sales of Child Car Seats")%>% 
             row_spec(row =0, bold= TRUE, color = "black", background = "#F9EBEA") %>%
             kable_styling(bootstrap_options = "striped", full_width = F, position = "center") %>%
             kable_paper() %>%
             scroll_box(width = "100%", height = "400px")
Sales of Child Car Seats
Sales CompPrice Income Advertising Population Price ShelveLoc Age Education Urban US
9.50 138 73 11 276 120 Bad 42 17 Yes Yes
11.22 111 48 16 260 83 Good 65 10 Yes Yes
10.06 113 35 10 269 80 Medium 59 12 Yes Yes
7.40 117 100 4 466 97 Medium 55 14 Yes Yes
4.15 141 64 3 340 128 Bad 38 13 Yes No
10.81 124 113 13 501 72 Bad 78 16 No Yes
6.63 115 105 0 45 108 Medium 71 15 Yes No
11.85 136 81 15 425 120 Good 67 10 Yes Yes
6.54 132 110 0 108 124 Medium 76 10 No No
4.69 132 113 0 131 124 Medium 76 17 No Yes
9.01 121 78 9 150 100 Bad 26 10 No Yes
11.96 117 94 4 503 94 Good 50 13 Yes Yes
3.98 122 35 2 393 136 Medium 62 18 Yes No
10.96 115 28 11 29 86 Good 53 18 Yes Yes
11.17 107 117 11 148 118 Good 52 18 Yes Yes
8.71 149 95 5 400 144 Medium 76 18 No No
7.58 118 32 0 284 110 Good 63 13 Yes No
12.29 147 74 13 251 131 Good 52 10 Yes Yes
13.91 110 110 0 408 68 Good 46 17 No Yes
8.73 129 76 16 58 121 Medium 69 12 Yes Yes
6.41 125 90 2 367 131 Medium 35 18 Yes Yes
12.13 134 29 12 239 109 Good 62 18 No Yes
5.08 128 46 6 497 138 Medium 42 13 Yes No
5.87 121 31 0 292 109 Medium 79 10 Yes No
10.14 145 119 16 294 113 Bad 42 12 Yes Yes
14.90 139 32 0 176 82 Good 54 11 No No
8.33 107 115 11 496 131 Good 50 11 No Yes
5.27 98 118 0 19 107 Medium 64 17 Yes No
2.99 103 74 0 359 97 Bad 55 11 Yes Yes
7.81 104 99 15 226 102 Bad 58 17 Yes Yes
13.55 125 94 0 447 89 Good 30 12 Yes No
8.25 136 58 16 241 131 Medium 44 18 Yes Yes
6.20 107 32 12 236 137 Good 64 10 No Yes
8.77 114 38 13 317 128 Good 50 16 Yes Yes
2.67 115 54 0 406 128 Medium 42 17 Yes Yes
11.07 131 84 11 29 96 Medium 44 17 No Yes
8.89 122 76 0 270 100 Good 60 18 No No
4.95 121 41 5 412 110 Medium 54 10 Yes Yes
6.59 109 73 0 454 102 Medium 65 15 Yes No
3.24 130 60 0 144 138 Bad 38 10 No No
2.07 119 98 0 18 126 Bad 73 17 No No
7.96 157 53 0 403 124 Bad 58 16 Yes No
10.43 77 69 0 25 24 Medium 50 18 Yes No
4.12 123 42 11 16 134 Medium 59 13 Yes Yes
4.16 85 79 6 325 95 Medium 69 13 Yes Yes
4.56 141 63 0 168 135 Bad 44 12 Yes Yes
12.44 127 90 14 16 70 Medium 48 15 No Yes
4.38 126 98 0 173 108 Bad 55 16 Yes No
3.91 116 52 0 349 98 Bad 69 18 Yes No
10.61 157 93 0 51 149 Good 32 17 Yes No
1.42 99 32 18 341 108 Bad 80 16 Yes Yes
4.42 121 90 0 150 108 Bad 75 16 Yes No
7.91 153 40 3 112 129 Bad 39 18 Yes Yes
6.92 109 64 13 39 119 Medium 61 17 Yes Yes
4.90 134 103 13 25 144 Medium 76 17 No Yes
6.85 143 81 5 60 154 Medium 61 18 Yes Yes
11.91 133 82 0 54 84 Medium 50 17 Yes No
0.91 93 91 0 22 117 Bad 75 11 Yes No
5.42 103 93 15 188 103 Bad 74 16 Yes Yes
5.21 118 71 4 148 114 Medium 80 13 Yes No
8.32 122 102 19 469 123 Bad 29 13 Yes Yes
7.32 105 32 0 358 107 Medium 26 13 No No
1.82 139 45 0 146 133 Bad 77 17 Yes Yes
8.47 119 88 10 170 101 Medium 61 13 Yes Yes
7.80 100 67 12 184 104 Medium 32 16 No Yes
4.90 122 26 0 197 128 Medium 55 13 No No
8.85 127 92 0 508 91 Medium 56 18 Yes No
9.01 126 61 14 152 115 Medium 47 16 Yes Yes
13.39 149 69 20 366 134 Good 60 13 Yes Yes
7.99 127 59 0 339 99 Medium 65 12 Yes No
9.46 89 81 15 237 99 Good 74 12 Yes Yes
6.50 148 51 16 148 150 Medium 58 17 No Yes
5.52 115 45 0 432 116 Medium 25 15 Yes No
12.61 118 90 10 54 104 Good 31 11 No Yes
6.20 150 68 5 125 136 Medium 64 13 No Yes
8.55 88 111 23 480 92 Bad 36 16 No Yes
10.64 102 87 10 346 70 Medium 64 15 Yes Yes
7.70 118 71 12 44 89 Medium 67 18 No Yes
4.43 134 48 1 139 145 Medium 65 12 Yes Yes
9.14 134 67 0 286 90 Bad 41 13 Yes No
8.01 113 100 16 353 79 Bad 68 11 Yes Yes
7.52 116 72 0 237 128 Good 70 13 Yes No
11.62 151 83 4 325 139 Good 28 17 Yes Yes
4.42 109 36 7 468 94 Bad 56 11 Yes Yes
2.23 111 25 0 52 121 Bad 43 18 No No
8.47 125 103 0 304 112 Medium 49 13 No No
8.70 150 84 9 432 134 Medium 64 15 Yes No
11.70 131 67 7 272 126 Good 54 16 No Yes
6.56 117 42 7 144 111 Medium 62 10 Yes Yes
7.95 128 66 3 493 119 Medium 45 16 No No
5.33 115 22 0 491 103 Medium 64 11 No No
4.81 97 46 11 267 107 Medium 80 15 Yes Yes
4.53 114 113 0 97 125 Medium 29 12 Yes No
8.86 145 30 0 67 104 Medium 55 17 Yes No
8.39 115 97 5 134 84 Bad 55 11 Yes Yes
5.58 134 25 10 237 148 Medium 59 13 Yes Yes
9.48 147 42 10 407 132 Good 73 16 No Yes
7.45 161 82 5 287 129 Bad 33 16 Yes Yes
12.49 122 77 24 382 127 Good 36 16 No Yes
4.88 121 47 3 220 107 Bad 56 16 No Yes
4.11 113 69 11 94 106 Medium 76 12 No Yes
6.20 128 93 0 89 118 Medium 34 18 Yes No
5.30 113 22 0 57 97 Medium 65 16 No No
5.07 123 91 0 334 96 Bad 78 17 Yes Yes
4.62 121 96 0 472 138 Medium 51 12 Yes No
5.55 104 100 8 398 97 Medium 61 11 Yes Yes
0.16 102 33 0 217 139 Medium 70 18 No No
8.55 134 107 0 104 108 Medium 60 12 Yes No
3.47 107 79 2 488 103 Bad 65 16 Yes No
8.98 115 65 0 217 90 Medium 60 17 No No
9.00 128 62 7 125 116 Medium 43 14 Yes Yes
6.62 132 118 12 272 151 Medium 43 14 Yes Yes
6.67 116 99 5 298 125 Good 62 12 Yes Yes
6.01 131 29 11 335 127 Bad 33 12 Yes Yes
9.31 122 87 9 17 106 Medium 65 13 Yes Yes
8.54 139 35 0 95 129 Medium 42 13 Yes No
5.08 135 75 0 202 128 Medium 80 10 No No
8.80 145 53 0 507 119 Medium 41 12 Yes No
7.57 112 88 2 243 99 Medium 62 11 Yes Yes
7.37 130 94 8 137 128 Medium 64 12 Yes Yes
6.87 128 105 11 249 131 Medium 63 13 Yes Yes
11.67 125 89 10 380 87 Bad 28 10 Yes Yes
6.88 119 100 5 45 108 Medium 75 10 Yes Yes
8.19 127 103 0 125 155 Good 29 15 No Yes
8.87 131 113 0 181 120 Good 63 14 Yes No
9.34 89 78 0 181 49 Medium 43 15 No No
11.27 153 68 2 60 133 Good 59 16 Yes Yes
6.52 125 48 3 192 116 Medium 51 14 Yes Yes
4.96 133 100 3 350 126 Bad 55 13 Yes Yes
4.47 143 120 7 279 147 Bad 40 10 No Yes
8.41 94 84 13 497 77 Medium 51 12 Yes Yes
6.50 108 69 3 208 94 Medium 77 16 Yes No
9.54 125 87 9 232 136 Good 72 10 Yes Yes
7.62 132 98 2 265 97 Bad 62 12 Yes Yes
3.67 132 31 0 327 131 Medium 76 16 Yes No
6.44 96 94 14 384 120 Medium 36 18 No Yes
5.17 131 75 0 10 120 Bad 31 18 No No
6.52 128 42 0 436 118 Medium 80 11 Yes No
10.27 125 103 12 371 109 Medium 44 10 Yes Yes
12.30 146 62 10 310 94 Medium 30 13 No Yes
6.03 133 60 10 277 129 Medium 45 18 Yes Yes
6.53 140 42 0 331 131 Bad 28 15 Yes No
7.44 124 84 0 300 104 Medium 77 15 Yes No
0.53 122 88 7 36 159 Bad 28 17 Yes Yes
9.09 132 68 0 264 123 Good 34 11 No No
8.77 144 63 11 27 117 Medium 47 17 Yes Yes
3.90 114 83 0 412 131 Bad 39 14 Yes No
10.51 140 54 9 402 119 Good 41 16 No Yes
7.56 110 119 0 384 97 Medium 72 14 No Yes
11.48 121 120 13 140 87 Medium 56 11 Yes Yes
10.49 122 84 8 176 114 Good 57 10 No Yes
10.77 111 58 17 407 103 Good 75 17 No Yes
7.64 128 78 0 341 128 Good 45 13 No No
5.93 150 36 7 488 150 Medium 25 17 No Yes
6.89 129 69 10 289 110 Medium 50 16 No Yes
7.71 98 72 0 59 69 Medium 65 16 Yes No
7.49 146 34 0 220 157 Good 51 16 Yes No
10.21 121 58 8 249 90 Medium 48 13 No Yes
12.53 142 90 1 189 112 Good 39 10 No Yes
9.32 119 60 0 372 70 Bad 30 18 No No
4.67 111 28 0 486 111 Medium 29 12 No No
2.93 143 21 5 81 160 Medium 67 12 No Yes
3.63 122 74 0 424 149 Medium 51 13 Yes No
5.68 130 64 0 40 106 Bad 39 17 No No
8.22 148 64 0 58 141 Medium 27 13 No Yes
0.37 147 58 7 100 191 Bad 27 15 Yes Yes
6.71 119 67 17 151 137 Medium 55 11 Yes Yes
6.71 106 73 0 216 93 Medium 60 13 Yes No
7.30 129 89 0 425 117 Medium 45 10 Yes No
11.48 104 41 15 492 77 Good 73 18 Yes Yes
8.01 128 39 12 356 118 Medium 71 10 Yes Yes
12.49 93 106 12 416 55 Medium 75 15 Yes Yes
9.03 104 102 13 123 110 Good 35 16 Yes Yes
6.38 135 91 5 207 128 Medium 66 18 Yes Yes
0.00 139 24 0 358 185 Medium 79 15 No No
7.54 115 89 0 38 122 Medium 25 12 Yes No
5.61 138 107 9 480 154 Medium 47 11 No Yes
10.48 138 72 0 148 94 Medium 27 17 Yes Yes
10.66 104 71 14 89 81 Medium 25 14 No Yes
7.78 144 25 3 70 116 Medium 77 18 Yes Yes
4.94 137 112 15 434 149 Bad 66 13 Yes Yes
7.43 121 83 0 79 91 Medium 68 11 Yes No
4.74 137 60 4 230 140 Bad 25 13 Yes No
5.32 118 74 6 426 102 Medium 80 18 Yes Yes
9.95 132 33 7 35 97 Medium 60 11 No Yes
10.07 130 100 11 449 107 Medium 64 10 Yes Yes
8.68 120 51 0 93 86 Medium 46 17 No No
6.03 117 32 0 142 96 Bad 62 17 Yes No
8.07 116 37 0 426 90 Medium 76 15 Yes No
12.11 118 117 18 509 104 Medium 26 15 No Yes
8.79 130 37 13 297 101 Medium 37 13 No Yes
6.67 156 42 13 170 173 Good 74 14 Yes Yes
7.56 108 26 0 408 93 Medium 56 14 No No
13.28 139 70 7 71 96 Good 61 10 Yes Yes
7.23 112 98 18 481 128 Medium 45 11 Yes Yes
4.19 117 93 4 420 112 Bad 66 11 Yes Yes
4.10 130 28 6 410 133 Bad 72 16 Yes Yes
2.52 124 61 0 333 138 Medium 76 16 Yes No
3.62 112 80 5 500 128 Medium 69 10 Yes Yes
6.42 122 88 5 335 126 Medium 64 14 Yes Yes
5.56 144 92 0 349 146 Medium 62 12 No No
5.94 138 83 0 139 134 Medium 54 18 Yes No
4.10 121 78 4 413 130 Bad 46 10 No Yes
2.05 131 82 0 132 157 Bad 25 14 Yes No
8.74 155 80 0 237 124 Medium 37 14 Yes No
5.68 113 22 1 317 132 Medium 28 12 Yes No
4.97 162 67 0 27 160 Medium 77 17 Yes Yes
8.19 111 105 0 466 97 Bad 61 10 No No
7.78 86 54 0 497 64 Bad 33 12 Yes No
3.02 98 21 11 326 90 Bad 76 11 No Yes
4.36 125 41 2 357 123 Bad 47 14 No Yes
9.39 117 118 14 445 120 Medium 32 15 Yes Yes
12.04 145 69 19 501 105 Medium 45 11 Yes Yes
8.23 149 84 5 220 139 Medium 33 10 Yes Yes
4.83 115 115 3 48 107 Medium 73 18 Yes Yes
2.34 116 83 15 170 144 Bad 71 11 Yes Yes
5.73 141 33 0 243 144 Medium 34 17 Yes No
4.34 106 44 0 481 111 Medium 70 14 No No
9.70 138 61 12 156 120 Medium 25 14 Yes Yes
10.62 116 79 19 359 116 Good 58 17 Yes Yes
10.59 131 120 15 262 124 Medium 30 10 Yes Yes
6.43 124 44 0 125 107 Medium 80 11 Yes No
7.49 136 119 6 178 145 Medium 35 13 Yes Yes
3.45 110 45 9 276 125 Medium 62 14 Yes Yes
4.10 134 82 0 464 141 Medium 48 13 No No
6.68 107 25 0 412 82 Bad 36 14 Yes No
7.80 119 33 0 245 122 Good 56 14 Yes No
8.69 113 64 10 68 101 Medium 57 16 Yes Yes
5.40 149 73 13 381 163 Bad 26 11 No Yes
11.19 98 104 0 404 72 Medium 27 18 No No
5.16 115 60 0 119 114 Bad 38 14 No No
8.09 132 69 0 123 122 Medium 27 11 No No
13.14 137 80 10 24 105 Good 61 15 Yes Yes
8.65 123 76 18 218 120 Medium 29 14 No Yes
9.43 115 62 11 289 129 Good 56 16 No Yes
5.53 126 32 8 95 132 Medium 50 17 Yes Yes
9.32 141 34 16 361 108 Medium 69 10 Yes Yes
9.62 151 28 8 499 135 Medium 48 10 Yes Yes
7.36 121 24 0 200 133 Good 73 13 Yes No
3.89 123 105 0 149 118 Bad 62 16 Yes Yes
10.31 159 80 0 362 121 Medium 26 18 Yes No
12.01 136 63 0 160 94 Medium 38 12 Yes No
4.68 124 46 0 199 135 Medium 52 14 No No
7.82 124 25 13 87 110 Medium 57 10 Yes Yes
8.78 130 30 0 391 100 Medium 26 18 Yes No
10.00 114 43 0 199 88 Good 57 10 No Yes
6.90 120 56 20 266 90 Bad 78 18 Yes Yes
5.04 123 114 0 298 151 Bad 34 16 Yes No
5.36 111 52 0 12 101 Medium 61 11 Yes Yes
5.05 125 67 0 86 117 Bad 65 11 Yes No
9.16 137 105 10 435 156 Good 72 14 Yes Yes
3.72 139 111 5 310 132 Bad 62 13 Yes Yes
8.31 133 97 0 70 117 Medium 32 16 Yes No
5.64 124 24 5 288 122 Medium 57 12 No Yes
9.58 108 104 23 353 129 Good 37 17 Yes Yes
7.71 123 81 8 198 81 Bad 80 15 Yes Yes
4.20 147 40 0 277 144 Medium 73 10 Yes No
8.67 125 62 14 477 112 Medium 80 13 Yes Yes
3.47 108 38 0 251 81 Bad 72 14 No No
5.12 123 36 10 467 100 Bad 74 11 No Yes
7.67 129 117 8 400 101 Bad 36 10 Yes Yes
5.71 121 42 4 188 118 Medium 54 15 Yes Yes
6.37 120 77 15 86 132 Medium 48 18 Yes Yes
7.77 116 26 6 434 115 Medium 25 17 Yes Yes
6.95 128 29 5 324 159 Good 31 15 Yes Yes
5.31 130 35 10 402 129 Bad 39 17 Yes Yes
9.10 128 93 12 343 112 Good 73 17 No Yes
5.83 134 82 7 473 112 Bad 51 12 No Yes
6.53 123 57 0 66 105 Medium 39 11 Yes No
5.01 159 69 0 438 166 Medium 46 17 Yes No
11.99 119 26 0 284 89 Good 26 10 Yes No
4.55 111 56 0 504 110 Medium 62 16 Yes No
12.98 113 33 0 14 63 Good 38 12 Yes No
10.04 116 106 8 244 86 Medium 58 12 Yes Yes
7.22 135 93 2 67 119 Medium 34 11 Yes Yes
6.67 107 119 11 210 132 Medium 53 11 Yes Yes
6.93 135 69 14 296 130 Medium 73 15 Yes Yes
7.80 136 48 12 326 125 Medium 36 16 Yes Yes
7.22 114 113 2 129 151 Good 40 15 No Yes
3.42 141 57 13 376 158 Medium 64 18 Yes Yes
2.86 121 86 10 496 145 Bad 51 10 Yes Yes
11.19 122 69 7 303 105 Good 45 16 No Yes
7.74 150 96 0 80 154 Good 61 11 Yes No
5.36 135 110 0 112 117 Medium 80 16 No No
6.97 106 46 11 414 96 Bad 79 17 No No
7.60 146 26 11 261 131 Medium 39 10 Yes Yes
7.53 117 118 11 429 113 Medium 67 18 No Yes
6.88 95 44 4 208 72 Bad 44 17 Yes Yes
6.98 116 40 0 74 97 Medium 76 15 No No
8.75 143 77 25 448 156 Medium 43 17 Yes Yes
9.49 107 111 14 400 103 Medium 41 11 No Yes
6.64 118 70 0 106 89 Bad 39 17 Yes No
11.82 113 66 16 322 74 Good 76 15 Yes Yes
11.28 123 84 0 74 89 Good 59 10 Yes No
12.66 148 76 3 126 99 Good 60 11 Yes Yes
4.21 118 35 14 502 137 Medium 79 10 No Yes
8.21 127 44 13 160 123 Good 63 18 Yes Yes
3.07 118 83 13 276 104 Bad 75 10 Yes Yes
10.98 148 63 0 312 130 Good 63 15 Yes No
9.40 135 40 17 497 96 Medium 54 17 No Yes
8.57 116 78 1 158 99 Medium 45 11 Yes Yes
7.41 99 93 0 198 87 Medium 57 16 Yes Yes
5.28 108 77 13 388 110 Bad 74 14 Yes Yes
10.01 133 52 16 290 99 Medium 43 11 Yes Yes
11.93 123 98 12 408 134 Good 29 10 Yes Yes
8.03 115 29 26 394 132 Medium 33 13 Yes Yes
4.78 131 32 1 85 133 Medium 48 12 Yes Yes
5.90 138 92 0 13 120 Bad 61 12 Yes No
9.24 126 80 19 436 126 Medium 52 10 Yes Yes
11.18 131 111 13 33 80 Bad 68 18 Yes Yes
9.53 175 65 29 419 166 Medium 53 12 Yes Yes
6.15 146 68 12 328 132 Bad 51 14 Yes Yes
6.80 137 117 5 337 135 Bad 38 10 Yes Yes
9.33 103 81 3 491 54 Medium 66 13 Yes No
7.72 133 33 10 333 129 Good 71 14 Yes Yes
6.39 131 21 8 220 171 Good 29 14 Yes Yes
15.63 122 36 5 369 72 Good 35 10 Yes Yes
6.41 142 30 0 472 136 Good 80 15 No No
10.08 116 72 10 456 130 Good 41 14 No Yes
6.97 127 45 19 459 129 Medium 57 11 No Yes
5.86 136 70 12 171 152 Medium 44 18 Yes Yes
7.52 123 39 5 499 98 Medium 34 15 Yes No
9.16 140 50 10 300 139 Good 60 15 Yes Yes
10.36 107 105 18 428 103 Medium 34 12 Yes Yes
2.66 136 65 4 133 150 Bad 53 13 Yes Yes
11.70 144 69 11 131 104 Medium 47 11 Yes Yes
4.69 133 30 0 152 122 Medium 53 17 Yes No
6.23 112 38 17 316 104 Medium 80 16 Yes Yes
3.15 117 66 1 65 111 Bad 55 11 Yes Yes
11.27 100 54 9 433 89 Good 45 12 Yes Yes
4.99 122 59 0 501 112 Bad 32 14 No No
10.10 135 63 15 213 134 Medium 32 10 Yes Yes
5.74 106 33 20 354 104 Medium 61 12 Yes Yes
5.87 136 60 7 303 147 Medium 41 10 Yes Yes
7.63 93 117 9 489 83 Bad 42 13 Yes Yes
6.18 120 70 15 464 110 Medium 72 15 Yes Yes
5.17 138 35 6 60 143 Bad 28 18 Yes No
8.61 130 38 0 283 102 Medium 80 15 Yes No
5.97 112 24 0 164 101 Medium 45 11 Yes No
11.54 134 44 4 219 126 Good 44 15 Yes Yes
7.50 140 29 0 105 91 Bad 43 16 Yes No
7.38 98 120 0 268 93 Medium 72 10 No No
7.81 137 102 13 422 118 Medium 71 10 No Yes
5.99 117 42 10 371 121 Bad 26 14 Yes Yes
8.43 138 80 0 108 126 Good 70 13 No Yes
4.81 121 68 0 279 149 Good 79 12 Yes No
8.97 132 107 0 144 125 Medium 33 13 No No
6.88 96 39 0 161 112 Good 27 14 No No
12.57 132 102 20 459 107 Good 49 11 Yes Yes
9.32 134 27 18 467 96 Medium 49 14 No Yes
8.64 111 101 17 266 91 Medium 63 17 No Yes
10.44 124 115 16 458 105 Medium 62 16 No Yes
13.44 133 103 14 288 122 Good 61 17 Yes Yes
9.45 107 67 12 430 92 Medium 35 12 No Yes
5.30 133 31 1 80 145 Medium 42 18 Yes Yes
7.02 130 100 0 306 146 Good 42 11 Yes No
3.58 142 109 0 111 164 Good 72 12 Yes No
13.36 103 73 3 276 72 Medium 34 15 Yes Yes
4.17 123 96 10 71 118 Bad 69 11 Yes Yes
3.13 130 62 11 396 130 Bad 66 14 Yes Yes
8.77 118 86 7 265 114 Good 52 15 No Yes
8.68 131 25 10 183 104 Medium 56 15 No Yes
5.25 131 55 0 26 110 Bad 79 12 Yes Yes
10.26 111 75 1 377 108 Good 25 12 Yes No
10.50 122 21 16 488 131 Good 30 14 Yes Yes
6.53 154 30 0 122 162 Medium 57 17 No No
5.98 124 56 11 447 134 Medium 53 12 No Yes
14.37 95 106 0 256 53 Good 52 17 Yes No
10.71 109 22 10 348 79 Good 74 14 No Yes
10.26 135 100 22 463 122 Medium 36 14 Yes Yes
7.68 126 41 22 403 119 Bad 42 12 Yes Yes
9.08 152 81 0 191 126 Medium 54 16 Yes No
7.80 121 50 0 508 98 Medium 65 11 No No
5.58 137 71 0 402 116 Medium 78 17 Yes No
9.44 131 47 7 90 118 Medium 47 12 Yes Yes
7.90 132 46 4 206 124 Medium 73 11 Yes No
16.27 141 60 19 319 92 Good 44 11 Yes Yes
6.81 132 61 0 263 125 Medium 41 12 No No
6.11 133 88 3 105 119 Medium 79 12 Yes Yes
5.81 125 111 0 404 107 Bad 54 15 Yes No
9.64 106 64 10 17 89 Medium 68 17 Yes Yes
3.90 124 65 21 496 151 Bad 77 13 Yes Yes
4.95 121 28 19 315 121 Medium 66 14 Yes Yes
9.35 98 117 0 76 68 Medium 63 10 Yes No
12.85 123 37 15 348 112 Good 28 12 Yes Yes
5.87 131 73 13 455 132 Medium 62 17 Yes Yes
5.32 152 116 0 170 160 Medium 39 16 Yes No
8.67 142 73 14 238 115 Medium 73 14 No Yes
8.14 135 89 11 245 78 Bad 79 16 Yes Yes
8.44 128 42 8 328 107 Medium 35 12 Yes Yes
5.47 108 75 9 61 111 Medium 67 12 Yes Yes
6.10 153 63 0 49 124 Bad 56 16 Yes No
4.53 129 42 13 315 130 Bad 34 13 Yes Yes
5.57 109 51 10 26 120 Medium 30 17 No Yes
5.35 130 58 19 366 139 Bad 33 16 Yes Yes
12.57 138 108 17 203 128 Good 33 14 Yes Yes
6.14 139 23 3 37 120 Medium 55 11 No Yes
7.41 162 26 12 368 159 Medium 40 18 Yes Yes
5.94 100 79 7 284 95 Bad 50 12 Yes Yes
9.71 134 37 0 27 120 Good 49 16 Yes Yes

a. Fit a multiple regression model to predict Sales using Price, Urban and US.

model_2 <- lm(Sales ~ Price + Urban + US, data = dat)
summary(model_2)
## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9206 -1.6220 -0.0564  1.5786  7.0581 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
## Price       -0.054459   0.005242 -10.389  < 2e-16 ***
## UrbanYes    -0.021916   0.271650  -0.081    0.936    
## USYes        1.200573   0.259042   4.635 4.86e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2335 
## F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

b. Provide an interpretation of each coefficient in the model. Be careful–some of the variables in the model are qualitative!

model_2$coefficients
## (Intercept)       Price    UrbanYes       USYes 
## 13.04346894 -0.05445885 -0.02191615  1.20057270

In the model, the interpretation of each coefficient indicates that: +) an unit increases in Price will decreases 0.052 unit in Sales +) a choice of Yes in Urban will decreases 0.062 unit in Sales +) a choice of Yes in US will decreases 1.052 unit in Sales

Note that: Urban and US, are represented in the model as dummy variables (binary indicators), where the value of 1 indicates the presence of the characteristic (e.g. urban) and 0 indicates the absence of the characteristic (e.g. rural or non-US).

c. Write out the model in equation form, being careful to handle the qualitative variables properly.

x <- as.data.frame(model_2$coefficients, col.names = "intercof") %>% format(digits = 3)
print(paste("Sales = ",x[1,1],"+", x[2,1], "x Price +", x[3,1],"x Urban +", x[4,1], "x US"))
## [1] "Sales =  13.0435 + -0.0545 x Price + -0.0219 x Urban +  1.2006 x US"

d. For which of the predictors can you reject the null hypothesis 𝐻0:𝛽𝑗=0?

summary(model_2)
## 
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9206 -1.6220 -0.0564  1.5786  7.0581 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.043469   0.651012  20.036  < 2e-16 ***
## Price       -0.054459   0.005242 -10.389  < 2e-16 ***
## UrbanYes    -0.021916   0.271650  -0.081    0.936    
## USYes        1.200573   0.259042   4.635 4.86e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2335 
## F-statistic: 41.52 on 3 and 396 DF,  p-value: < 2.2e-16

From the result, with the p-value < 0.05, Price and US are 2 significant independent variables in the model which can be rejected for the null hypothesis of being 0 .

e. On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome.

#We take the 2 significant variables Price and US to conduct new small model
model_3 <- lm(Sales ~ Price + US, data = dat)
summary(model_3)
## 
## Call:
## lm(formula = Sales ~ Price + US, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9269 -1.6286 -0.0574  1.5766  7.0515 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.03079    0.63098  20.652  < 2e-16 ***
## Price       -0.05448    0.00523 -10.416  < 2e-16 ***
## USYes        1.19964    0.25846   4.641 4.71e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2354 
## F-statistic: 62.43 on 2 and 397 DF,  p-value: < 2.2e-16

f. How well do the models in (a) and (e) fit the data?

To compare the result of 2 models, we will create a Root Mean Square Error function to test the data:

library(caret)
## Loading required package: lattice
## 
## Attaching package: 'caret'
## The following objects are masked from 'package:yardstick':
## 
##     precision, recall, sensitivity, specificity
## The following object is masked from 'package:purrr':
## 
##     lift
library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:ISLR2':
## 
##     Boston
## The following object is masked from 'package:dplyr':
## 
##     select
#RMSE
RMSE <- function(x,y){
  return(sqrt(mean((x-y)^2)))
}
test_index <- createDataPartition(dat$Sales, times = 1, p = 0.2, list = FALSE)

#Randomly divide dataset into 80% train data and 20% test data
set.seed(1)
train <- dat[-test_index,]
test <- dat[test_index,]
dim(train)
## [1] 319  11
dim(test)
## [1] 81 11

Then, we will find the model from the train data and apply the model into test data:

#For the model in question (a) which includes Price, US and Urban: 
model_a <- lm(Sales ~ Price + Urban + US, data = train)
predict <- predict(model_a, newdata = test)
RMSE(predict,test$Sales)
## [1] 2.60792
#For the model in question (e) which includes Price, US: 
model_e <- lm(Sales ~ Price + US, data = train)
predict <- predict(model_e, newdata = test)
RMSE(predict,test$Sales)
## [1] 2.600727

The Root Mean Square Error in the model in question (e) is smaller the the one in question(a), which means that the new model including Price and US is fitter and more correct for the data.

g. Using the model from (e), obtain 95% confidence intervals for the coefficient(s).

summary(model_3)
## 
## Call:
## lm(formula = Sales ~ Price + US, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9269 -1.6286 -0.0574  1.5766  7.0515 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13.03079    0.63098  20.652  < 2e-16 ***
## Price       -0.05448    0.00523 -10.416  < 2e-16 ***
## USYes        1.19964    0.25846   4.641 4.71e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared:  0.2393, Adjusted R-squared:  0.2354 
## F-statistic: 62.43 on 2 and 397 DF,  p-value: < 2.2e-16

The p-value of Price and US are lower than 0.05, which obtain the 95% confidence.

h. Is there evidence of outliers or high leverage observations in the model from (e)?

To detect the evidence of outliers or high leverage observations in the model from (e), we apply the Cook’s distance measures:

par(mfrow = c(2, 2))
plot(model_3)

From the plot, we do not have any outliner nor high levarage observation compared to the Cook’s distance.