Simple linear regression is a statistical measure used to examine the relationship between two numerical values.

• Independent variable (X): : Vehicle weight
• Dependent variable (Y): Fuel efficiency (MPG)
• Goal: Determine whether vehicle weight can predict fuel efficiency.

For this analysis, we will use built-in mtcars dataset, which contains information about 32 vehicles.

Simple linear regression uses the following equation to predict an outcome:

\[ \hat{Y} = \beta_0 + \beta_1 X \]

Where:

• Y-hat: Predicted value of the dependent variable
• Beta 0: Intercept, or predicted Y when X = 0
• Beta 1: Slope, or change in predicted Y for each one-unit increase in X
• X: Independent variable

In our analysis, we will use vehicle weight

The mtcars dataset contains information about 32 vehicles.

We will focus on two variables:

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

Our goal is to determine whether heavier vehicles tend to have lower fuel efficiency.

##                      wt  mpg
## Mazda RX4         2.620 21.0
## Mazda RX4 Wag     2.875 21.0
## Datsun 710        2.320 22.8
## Hornet 4 Drive    3.215 21.4
## Hornet Sportabout 3.440 18.7
## Valiant           3.460 18.1

This scatterplot illustrates the relationship between vehicle weight and fuel efficiency.

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

The fitted regression equation is: \[\widehat{MPG} = 37.2851 - 5.3445(wt)\] Where \(wt\) represents vehicle weight in thousands of pounds.

## 
## 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

This histogram illustrates the distribution of vehicle weights among the 32 vehicles in the mtcars dataset. Vehicle weight is measured in thousands of pounds.

The regression analysis identified a strong negative relationship between vehicle weight and fuel efficiency.

• Regression equation: MPG = 37.2851 - 5.3445(WT)

• Slope: Every additional 1,000 pounds of vehicle weight is associated with an estimated 5.34 MPG decrese.

• R-squared: Vehicle weight explains approximately 75.28% of the variation in MPG.

• P-value: 1.29 x 10^(-10), indicating a statistically significant relationship.

Conclusion: Heavier vehicles gnerally have lower fuel efficiency. Vehicle weight is a strong predictor of MPG in the mtcars dataset.

This interactive scatterplot shows the relationship between vehicle weight and fuel efficiency.

Our simple linear regression analysis found a strong negative relationship between vehicle weight and fuel efficiency.

• Regression equation: MPG = 37.285 -5.344(wt)
• Slope: For every additional 1,000 pounds of vehicle weight, predicted fuel efficiency decreases by approximately 5.34 MPG.
• R-squared: 0.7528, meaning vehicle weight explains approximately 75.3% of the variation in fuel efficiency.
• P-value: 1.29e-10, indicating a statistically significant relationship.

Overall, heavier vehicles tend to have lower fuel efficiency. Vehicle weight is a strong predictor of the MPG in the mtcars dataset.

The following R code creates our regression model:

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