Can vehicle weight predict fuel efficiency?

Simple linear regression helps us understand the relationship between two numerical variables.

In this presentation, we will explore whether heavier vehicles tend to have lower fuel efficiency using the built-in mtcars dataset.

The mtcars dataset contains information about 32 vehicles.

We will focus on two variables:

• wt: Vehicle weight in thousands of pounds.
• mpg: Fuel efficiency in miles per gallon.

library(ggplot2)
library(plotly)
library(dplyr)

data(mtcars)
head(mtcars)

##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Simple linear regression models the relationship between two numerical variables.

The equation is:

\[ Y = \beta_0 + \beta_1 X + \epsilon \]

Where:

• \(Y\) = Dependent variable (fuel efficiency)
• \(X\) = Independent variable (vehicle weight)
• \(\beta_0\) = Intercept
• \(\beta_1\) = Slope
• \(\epsilon\) = Random error

The goal is to estimate how changes in vehicle weight affect fuel efficiency.

We can estimate the relationship between weight and fuel efficiency using R’s lm() function.

model <- lm(mpg ~ wt, data = mtcars)

summary(model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

The regression model estimates how much fuel efficiency changes for every additional 1,000 pounds of vehicle weight.

The fitted regression equation is:

\[ \widehat{MPG} = 37.285 - 5.344(WT) \]

For a vehicle weighing 3,000 pounds, \(WT = 3\):

\[ \widehat{MPG} = 37.285 - 5.344(3) \]

\[ \widehat{MPG} \approx 21.25 \]

Interpretation: A vehicle weighing 3,000 pounds is predicted to achieve approximately 21.25 miles per gallon.

## `geom_smooth()` using formula = 'y ~ x'

The downward trend suggests that heavier vehicles generally have lower fuel efficiency.

Residuals are the differences between actual and predicted MPG. Examining residuals helps us evaluate how well the regression model fits the data.

Residuals should ideally be scattered around zero without a clear pattern.

This interactive visualization allows users to explore vehicle weight and fuel efficiency.

Key Findings

• Heavier vehicles generally have lower fuel efficiency.
• Each additional 1,000 pounds of vehicle weight is associated with approximately 5.34 fewer miles per gallon.
• A vehicle weighing 3,000 pounds is predicted to achieve approximately 21.25 MPG.
• Simple linear regression provides a useful way to understand relationships and make predictions.

Important: Correlation and regression do not establish causation.