Assignment3

Libraries

library(tidyverse)

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## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
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library(ggplot2)
library(DescTools)

Data Exploration

Load the mtcars data set, review its code book, and use summary statics and visualizations to understand the data set.

help(mtcars)
mtcars

##                      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

summary(mtcars)

##       mpg             cyl             disp             hp       
##  Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
##  1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
##  Median :19.20   Median :6.000   Median :196.3   Median :123.0  
##  Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
##  3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
##  Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
##       drat             wt             qsec             vs        
##  Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
##  1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
##  Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
##  Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
##  3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
##  Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
##        am              gear            carb      
##  Min.   :0.0000   Min.   :3.000   Min.   :1.000  
##  1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
##  Median :0.0000   Median :4.000   Median :2.000  
##  Mean   :0.4062   Mean   :3.688   Mean   :2.812  
##  3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
##  Max.   :1.0000   Max.   :5.000   Max.   :8.000

ncol(mtcars)

## [1] 11

nrow(mtcars)

## [1] 32

Data Exploration

Identify at least one trends, correlations, or patterns you find interesting!

I found the correlation between weight and mpg intresting because I knew that the more a car weighs, the lower gas milage it will have. So it was cool to be able to see that fact in graph form.

ggplot(mtcars, aes(x=wt, y=mpg)) +
  geom_point()

What variables are most strongly correlated with mpg? Why?

The variable that is most strongly correlated with mpg is wt with a value of -0.867. The second closest value is cyl with a value of -0.852

data(mtcars)
cor_value <- cor(mtcars)
print(cor_value)

##             mpg        cyl       disp         hp        drat         wt
## mpg   1.0000000 -0.8521620 -0.8475514 -0.7761684  0.68117191 -0.8676594
## cyl  -0.8521620  1.0000000  0.9020329  0.8324475 -0.69993811  0.7824958
## disp -0.8475514  0.9020329  1.0000000  0.7909486 -0.71021393  0.8879799
## hp   -0.7761684  0.8324475  0.7909486  1.0000000 -0.44875912  0.6587479
## drat  0.6811719 -0.6999381 -0.7102139 -0.4487591  1.00000000 -0.7124406
## wt   -0.8676594  0.7824958  0.8879799  0.6587479 -0.71244065  1.0000000
## qsec  0.4186840 -0.5912421 -0.4336979 -0.7082234  0.09120476 -0.1747159
## vs    0.6640389 -0.8108118 -0.7104159 -0.7230967  0.44027846 -0.5549157
## am    0.5998324 -0.5226070 -0.5912270 -0.2432043  0.71271113 -0.6924953
## gear  0.4802848 -0.4926866 -0.5555692 -0.1257043  0.69961013 -0.5832870
## carb -0.5509251  0.5269883  0.3949769  0.7498125 -0.09078980  0.4276059
##             qsec         vs          am       gear        carb
## mpg   0.41868403  0.6640389  0.59983243  0.4802848 -0.55092507
## cyl  -0.59124207 -0.8108118 -0.52260705 -0.4926866  0.52698829
## disp -0.43369788 -0.7104159 -0.59122704 -0.5555692  0.39497686
## hp   -0.70822339 -0.7230967 -0.24320426 -0.1257043  0.74981247
## drat  0.09120476  0.4402785  0.71271113  0.6996101 -0.09078980
## wt   -0.17471588 -0.5549157 -0.69249526 -0.5832870  0.42760594
## qsec  1.00000000  0.7445354 -0.22986086 -0.2126822 -0.65624923
## vs    0.74453544  1.0000000  0.16834512  0.2060233 -0.56960714
## am   -0.22986086  0.1683451  1.00000000  0.7940588  0.05753435
## gear -0.21268223  0.2060233  0.79405876  1.0000000  0.27407284
## carb -0.65624923 -0.5696071  0.05753435  0.2740728  1.00000000

cor_value[, "mpg"]

##        mpg        cyl       disp         hp       drat         wt       qsec 
##  1.0000000 -0.8521620 -0.8475514 -0.7761684  0.6811719 -0.8676594  0.4186840 
##         vs         am       gear       carb 
##  0.6640389  0.5998324  0.4802848 -0.5509251

Data Processing

Check with there are missing data using R codes. If not, show evidence.

There are no missing values in the mtcars data set

is.na(mtcars)

##                       mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear
## Mazda RX4           FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Mazda RX4 Wag       FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Datsun 710          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Hornet 4 Drive      FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Hornet Sportabout   FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Valiant             FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Duster 360          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 240D           FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 230            FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 280            FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 280C           FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 450SE          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 450SL          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Merc 450SLC         FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Cadillac Fleetwood  FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Lincoln Continental FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Chrysler Imperial   FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Fiat 128            FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Honda Civic         FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Toyota Corolla      FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Toyota Corona       FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Dodge Challenger    FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## AMC Javelin         FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Camaro Z28          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Pontiac Firebird    FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Fiat X1-9           FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Porsche 914-2       FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Lotus Europa        FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Ford Pantera L      FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Ferrari Dino        FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Maserati Bora       FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## Volvo 142E          FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
##                      carb
## Mazda RX4           FALSE
## Mazda RX4 Wag       FALSE
## Datsun 710          FALSE
## Hornet 4 Drive      FALSE
## Hornet Sportabout   FALSE
## Valiant             FALSE
## Duster 360          FALSE
## Merc 240D           FALSE
## Merc 230            FALSE
## Merc 280            FALSE
## Merc 280C           FALSE
## Merc 450SE          FALSE
## Merc 450SL          FALSE
## Merc 450SLC         FALSE
## Cadillac Fleetwood  FALSE
## Lincoln Continental FALSE
## Chrysler Imperial   FALSE
## Fiat 128            FALSE
## Honda Civic         FALSE
## Toyota Corolla      FALSE
## Toyota Corona       FALSE
## Dodge Challenger    FALSE
## AMC Javelin         FALSE
## Camaro Z28          FALSE
## Pontiac Firebird    FALSE
## Fiat X1-9           FALSE
## Porsche 914-2       FALSE
## Lotus Europa        FALSE
## Ford Pantera L      FALSE
## Ferrari Dino        FALSE
## Maserati Bora       FALSE
## Volvo 142E          FALSE

sum(is.na(mtcars))

## [1] 0

Check whether there are inconsistent/invalid data. If not, show evidence.

After looking at the summary for the data set, we can see that there are no crazy outliars that skew the data

summary(mtcars)

##       mpg             cyl             disp             hp       
##  Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
##  1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
##  Median :19.20   Median :6.000   Median :196.3   Median :123.0  
##  Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
##  3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
##  Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
##       drat             wt             qsec             vs        
##  Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
##  1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
##  Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
##  Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
##  3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
##  Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
##        am              gear            carb      
##  Min.   :0.0000   Min.   :3.000   Min.   :1.000  
##  1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
##  Median :0.0000   Median :4.000   Median :2.000  
##  Mean   :0.4062   Mean   :3.688   Mean   :2.812  
##  3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
##  Max.   :1.0000   Max.   :5.000   Max.   :8.000

Linear Regression using lm

Create a linear regression model to predict mpg (miles per gallon) based on all the other variables in the dataset.

full_model <- lm(mpg ~ ., data = mtcars)
summary(full_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

Interpret the coefficients.

Based on the lm here, we are able to see again that wt is the most signifciant variable that affects mpg. It shows us that the heavier a car is, the lower the gas milage it will have. Other variables like hp and disp have such a low value that they don’t really affect mpg that much. With an R^2 value of .869, this means that 0.869% has been explained by the data. And an adjusted R-squared value of .806 tells us its an overall good fit

What assumptions are being made when we use linear regression? Are they met in this dataset? Just describe what you observe from the diagnostic plots.

The assumptions being made using linear regression is that the relationship between mpg and the other variables are linear and with multicollineraity, the variables are not too correlated with each other.

We can see when we use vif, that some variables have values over 10, like cyl, disp, and wt, that means that those variables are correlated which means that they have a high multicollineraity.

library(car)

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:DescTools':
## 
##     Recode

## The following object is masked from 'package:dplyr':
## 
##     recode

## The following object is masked from 'package:purrr':
## 
##     some

vif(full_model)

##       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

Evaluate the model using MSE (Mean Square Error)

predicted <- predict(full_model)
mse <- mean((mtcars$mpg - predicted)^2)
summary(mse)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   4.609   4.609   4.609   4.609   4.609   4.609

Try adding interaction terms to your linear regression model. At least try to find out one interaction term that has a statistically significant coefficient. Report the interaction term and check how do these interaction terms influence the model’s performance in terms of R^2 and how do you interpret your new model?

As we can see when we run this command, we can see that the affect of wt on am- it means that the wt on mpg is stronger than the affect of am on mpg.

The R^2 for this model compared to the full model is 0.869 compared to 0.9006 Out take away from these results is that the interaction model helps explain more of the variation in mpg

interaction <- lm(mpg ~ . + wt*am, data = mtcars)
summary(full_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

summary(interaction)

## 
## Call:
## lm(formula = mpg ~ . + wt * am, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0807 -1.4803 -0.4741  1.3226  4.5850 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  2.070807  17.191096   0.120  0.90532   
## cyl          0.225567   0.942201   0.239  0.81323   
## disp         0.004384   0.016328   0.269  0.79105   
## hp          -0.006131   0.020359  -0.301  0.76643   
## drat         0.359683   1.469370   0.245  0.80911   
## wt          -3.030663   1.712406  -1.770  0.09200 . 
## qsec         1.109647   0.662235   1.676  0.10938   
## vs          -0.077414   1.884792  -0.041  0.96764   
## am          13.334354   4.663324   2.859  0.00969 **
## gear         1.108383   1.344775   0.824  0.41954   
## carb        -0.090545   0.740919  -0.122  0.90395   
## wt:am       -4.137886   1.640318  -2.523  0.02023 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.365 on 20 degrees of freedom
## Multiple R-squared:  0.9006, Adjusted R-squared:  0.846 
## F-statistic: 16.48 on 11 and 20 DF,  p-value: 1.081e-07

Are there any outliers in the dataset? If yes, apply winsorization techniques to the variable with most outliers. Compare the performance of the models before and after applying these techniques. What differences do you observe in R^2 and the coefficients?

There are a few outliers. The biggest one is in hp, but wt, qsex, and carb also have outliers.

boxplot(mtcars, main="Boxplot for all variables")

winsorize <- mtcars
winsorize$hp <- Winsorize(mtcars$hp, val = quantile(mtcars$hp, probs = c(0.05, 0.95), na.rm = FALSE))

winsorize$wt <- Winsorize(mtcars$wt, val = quantile(mtcars$wt, probs = c(0.05, 0.95), na.rm = FALSE))

winsorize$qsec <- Winsorize(mtcars$qsec, val = quantile(mtcars$qsec, probs = c(0.05, 0.95), na.rm = FALSE))

winsorize$carb <- Winsorize(mtcars$carb, val = quantile(mtcars$carb, probs = c(0.05, 0.95), na.rm = FALSE))

summary(winsorize$hp)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   63.65   96.50  123.00  144.23  180.00  253.55

summary(winsorize$wt)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.736   2.581   3.325   3.222   3.610   5.293

summary(winsorize$qsec)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   15.05   16.89   17.71   17.79   18.90   20.10

summary(winsorize$carb)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   2.000   2.000   2.681   4.000   4.900

win_c <- winsorize [ , c("hp", "wt", "qsec", "carb")]
summary(win_c)

##        hp               wt             qsec            carb      
##  Min.   : 63.65   Min.   :1.736   Min.   :15.05   Min.   :1.000  
##  1st Qu.: 96.50   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:2.000  
##  Median :123.00   Median :3.325   Median :17.71   Median :2.000  
##  Mean   :144.23   Mean   :3.222   Mean   :17.79   Mean   :2.681  
##  3rd Qu.:180.00   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:4.000  
##  Max.   :253.55   Max.   :5.293   Max.   :20.10   Max.   :4.900

wins_model <- lm(mpg ~ ., data = winsorize)
summary(full_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

summary(wins_model)

## 
## Call:
## lm(formula = mpg ~ ., data = winsorize)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4747 -1.2929 -0.3913  0.9772  4.2880 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  8.41946   24.84625   0.339    0.738
## cyl         -0.20073    1.01372  -0.198    0.845
## disp         0.01124    0.01864   0.603    0.553
## hp          -0.02169    0.02658  -0.816    0.424
## drat         1.43716    1.85184   0.776    0.446
## wt          -3.46568    2.50599  -1.383    0.181
## qsec         0.97038    1.25256   0.775    0.447
## vs          -0.59474    2.52272  -0.236    0.816
## am           1.58236    1.99461   0.793    0.436
## gear         0.85737    1.44313   0.594    0.559
## carb        -0.50681    1.08940  -0.465    0.647
## 
## Residual standard error: 2.66 on 21 degrees of freedom
## Multiple R-squared:  0.868,  Adjusted R-squared:  0.8052 
## F-statistic: 13.81 on 10 and 21 DF,  p-value: 4.095e-07

Since out R^2 value went from 0.869 to 0.868, even though it was a very very small decrease, it means that our data better and more conscince. The small change makes sense becasue we did not have too many outliers in this data set, so the change being so small is expected.