Assignment 3
Tiernan Shibles
2025-10-21
Data Exploration
- Load the mtcars dataset, review its codebook, and use summary statistics and visualizations to understand the dataset.
## Loading in the dataset, reviewing the codebook, and using summary statistics
data(mtcars)
?mtcars
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## Visualizations
# Histogram of MPG
hist(mtcars$mpg, main = "Distribution of Miles Per Gallon", xlab = "MPG")
# Boxplot: MPG by Transmission Type
boxplot(mpg ~ am, data = mtcars,
names = c("Automatic", "Manual"),
main = "MPG by Transmission Type",
ylab = "Miles Per Gallon")
# Scatterplot: Weight vs MPG
plot(mtcars$wt, mtcars$mpg,
main = "MPG vs Weight",
xlab = "Weight (1000 lbs)",
ylab = "Miles Per Gallon")
- Identify at least one trend, correlation, or pattern you find interesting!
cor(mtcars$wt, mtcars$mpg)## [1] -0.8676594This answer means that heavier cars have a lower gallon per mileage, which makes sense because the heavier they the more fuel they have to use.
- What variables are most strongly correlated with mpg? Why?
cor(mtcars$mpg, mtcars) ## mpg cyl disp hp drat wt qsec
## [1,] 1 -0.852162 -0.8475514 -0.7761684 0.6811719 -0.8676594 0.418684
## vs am gear carb
## [1,] 0.6640389 0.5998324 0.4802848 -0.5509251Weight is most strongly correlated with MPG. The correlation is -0.868 which means that heavier cars require more fuel to move the same distance because of their weight. Secondly, cylinders and displacement have a correlation of about -0.85. This comes from the fact that more cylinders or larger engine displacement generally means that there is a bigger engine, so that engine burns more fuel.
Data Preprocessing
- Check whether there are missing data using R codes. If not, show evidence.
sum(is.na(mtcars))## [1] 0By running the sum of is.na in mtcars, it would return a number of misssing values in the dataset, and since there are no missing values, the number returned is 0.
- Check whether there are inconsistent/invalid data. If not, show evidence.
str(mtcars)## 'data.frame': 32 obs. of 11 variables:
## $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
## $ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
## $ disp: num 160 160 108 258 360 ...
## $ hp : num 110 110 93 110 175 105 245 62 95 123 ...
## $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
## $ wt : num 2.62 2.88 2.32 3.21 3.44 ...
## $ qsec: num 16.5 17 18.6 19.4 17 ...
## $ vs : num 0 0 1 1 0 1 0 1 1 1 ...
## $ am : num 1 1 1 0 0 0 0 0 0 0 ...
## $ gear: num 4 4 4 3 3 3 3 4 4 4 ...
## $ carb: num 4 4 1 1 2 1 4 2 2 4 ...## Confirms that all variables are numeric
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## All the variables summarized are logical in terms of their category
any(duplicated(mtcars))## [1] FALSE## The output of this is false showing that there are no duplicated values in the datasetLinear Regression using lm
- Create a linear regression model to predict mpg based on all the other variables in the dataset.
linear_model <- lm(mpg ~ ., data = mtcars)
summary(linear_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
- Intercept: 12.30, When all predictors are 0, the baseline mpg would be about 12.3
- Cyl: -0.11, Each additional cylinder slightly decreases mpg by 0.11
- Disp: 0.013, A 1 unit increase in engine displacement increases mpg by 0.013
- hp: -0.021, Each additional horsepower reduces mpg by about 0.02 miles per gallon, cars with higher horsepower use more fuel
- Drat: 0.79, A 1 unit increase in rear axle ratio is associated with a 0.79 increase in mpg.
- Wt: -3.72, For every additional 1000 pounds of car weight, mpg decreases by about 3.7 mpg.
- qsec: 0.82, Each additional second in the quarter-mile time increases mpg by 0.82. Slower cars are usually more efficient because they have smaller engines.
- vs: 0.32, Straight engines have slightly higher mpg than V-shaped engines, but it doesnt have much of an effect
- am: 2.52, Manual transmission cars get about 2.5 more miles per gallon that automatic ones
- gear, 0.66, Each additional gear slightly increases mpg by about 0.66.
- card, -0.20, Each additional carburetor slightly decreases mpg by 0.20
- What assumptions are being made when we use linear regression? Are they met in this dataset? The assumptions that are being made when doing linear regression are linearity, independence, homoscedasticity, normality of residuals, no multicollinearity.
par(mfrow = c(2,2))
plot(linear_model)
The linear regression assumptions are mostly met for the dataset, but multicollinearity between engine-related variables and slight-non linearity are noticeable. The model still performs well overall, but simplifying predictors would improve reliability.
- Evaluate the model using MSE.
MSE <- mean(linear_model$residuals^2)
MSE## [1] 4.609201- Adding interaction terms to your linear regression model.
model_interact <- lm(mpg ~ wt * am, data = mtcars)
summary(model_interact)##
## Call:
## lm(formula = mpg ~ wt * am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.6004 -1.5446 -0.5325 0.9012 6.0909
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.4161 3.0201 10.402 4.00e-11 ***
## wt -3.7859 0.7856 -4.819 4.55e-05 ***
## am 14.8784 4.2640 3.489 0.00162 **
## wt:am -5.2984 1.4447 -3.667 0.00102 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.591 on 28 degrees of freedom
## Multiple R-squared: 0.833, Adjusted R-squared: 0.8151
## F-statistic: 46.57 on 3 and 28 DF, p-value: 5.209e-11When adding an interaction term between weight and transmission type to the regression model, the interation was statistically significant. This means that the effect of a car’s weight on fuel efficiency (mpg) depends on whether the car has a manual or automatic transmission.
- Are there any outliers in the data. If so, winzorize the variable with the most outliers.
boxplot(mtcars, las = 2, main = "Boxplots for Outlier Detection")
## Horsepower and Displacement have several outliers, especially horsepower.boxplot.stats(mtcars$hp)$out## [1] 335## Checking to make sure that hp does have a good amount of outliers, and it does, so we will winsorize the variable.I was not sure how to winsorize the variable so I did not.