Simple Linear Regression models the relationship between one independent variable (\(x\)) and one dependent variable (\(y\)) using a straight line.

Ask a question? Is it possible to predict a car’s fuel efficiency (miles per gallon) from its weight?

I will use R’s mtcars dataset for this assignment.

The equation for simple linear regression is:

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

• \(Y\) = response variable (mpg)
• \(X\) = predictor variable (weight)
• \(\beta_0\) = intercept
• \(\beta_1\) = slope

model = lm(mpg ~ wt, data = mtcars)
summary(model)$coefficients

##              Estimate Std. Error   t value     Pr(>|t|)
## (Intercept) 37.285126   1.877627 19.857575 8.241799e-19
## wt          -5.344472   0.559101 -9.559044 1.293959e-10

• mtcars stores wt in thousands of pounds, e.g. 3,400 lbs has wt = 3.4.

From the model output:

\[\hat{Y} = 37.29 - 5.34 \cdot X\]

• The intercept is \(\hat{\beta}_0 = 37.29\)
• The slope is \(\hat{\beta}_1 = -5.34\)

This means for every 1000 lb increase in weight, mpg decreases by 5.3444.

The residual plot looks mostly random indicating a straight line is a good modeling method.

• Weight is a strong predictor of fuel efficiency.

summary(model)$r.squared

## [1] 0.7528328

• \(R^2 = 0.7528\), meaning 75% of the variation in mpg is explained by weight.
• The relationship is negative which indicates heavier cars get worse gas mileage than lighter ones.