• Simple linear regression studies the relationship between one predictor and one response.
• It helps us understand how changes in one variable are associated with changes in another.
• In this presentation:
  • Predictor: vehicle weight (wt)
  • Response: fuel efficiency (mpg)
• The goal is to fit a best-fit line to the data.

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

Where:

• \(Y\) = response variable, or the value we want to predict
  
• \(X\) = predictor variable
  
• \(\beta_0\) = intercept
  
• \(\beta_1\) = slope
  
• \(\varepsilon\) = random error

For this example:

\[ mpg = \beta_0 + \beta_1(wt) + \varepsilon \]

• I will use the built-in R dataset mtcars
• This dataset contains measurements for several automobiles
• For this example, I focus on:
  • wt = vehicle weight
  • mpg = miles per gallon
• Question:
  • Can vehicle weight help predict fuel efficiency?

\[ \hat{y} = b_0 + b_1x \]

\[ e_i = y_i - \hat{y}_i \]

Where:

• \(\hat{y}\) = predicted value from the regression line
• \(b_0\) = estimated intercept
• \(b_1\) = estimated slope
• \(e_i\) = residual, or prediction error
• \(y_i\) = actual observed value

For this example: - \(\hat{y}\) is the predicted mpg - the residual shows how far the real mpg is from the predicted mpg.

• Residuals are the differences between actual and predicted mpg values
• Most points are scattered around 0, which is a good sign
• There is no extremely strong pattern, so a linear model looks reasonable here.

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

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Fuel Efficiency vs Vehicle Weight",
    x = "Weight (1000 lbs)",
    y = "Miles Per Gallon"
  )