Simple linear regression is a method used to model the relationship between a dependent variable \(Y\) and an independent variable \(X\).

The equation for simple linear regression is:

\[ Y = \beta_0 + \beta_1 X + \epsilon \] Where: - \(Y\) is the dependent variable. - \(\beta_0\) is the intercept. - \(\beta_1\) is the slope. - \(X\) is the independent variable. - \(\epsilon\) is the error term.

The mtcars dataset contains specifications and performance data for 32 car models from the 1970s. The dataset includes 11 variables related to car performance, dimensions, and fuel consumption.

The data suggests there is an inverse relationship between weight and miles per gallon, meaning the heavier the car the less gas mileage.

Cars with the most horsepower tend to be heavier than cars with less horsepower.

Based off of the relationship between weight and miles per gallon, we will make a simple linear regression using the two variables.

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

model_summary <- summary(slr)

coefficients_rounded <- round(model_summary$coefficients, 3)

kable(coefficients_rounded, caption = "Regression Coefficients")

Diagnostic Plots of Simple Linear Regression

Diagnostic Plots of Multivariable Linear Regression

• Simple Linear Regression (SLR):
  \(\small \text{mpg} = 37.2851 - 5.3445 \times \text{wt}\)
  R-squared: 0.7528, explaining 75% of the variance in \(\small \text{mpg}\).
  p-value: \(\small < 2 \times 10^{-16}\), highly significant.

• Multivariable Linear Regression (MLR): \(\small \text{mpg} = 37.2273 - 3.8778 \times \text{wt} - 0.0318 \times \text{hp}\)
  R-squared: 0.8268, explaining 82.7% of the variance in \(\small \text{mpg}\).
  p-value: \(\small 9.109 \times 10^{-12}\), highly significant.

The MLR model explains 82.7% of the variance in mpg, compared to the 75% explained by the SLR model. Including horsepower (hp) improves the model’s fit, providing a higher R-squared. Both models are statistically significant, but the multivariable model offers a more robust prediction by accounting for weight (wt) and horsepower (hp). Therefore, for a more accurate prediction of mpg, the MLR model is recommended.