Topic: Simple linear regression
Question: How does car weight affect miles per gallon (mpg)?

We will use the built-in mtcars dataset and fit the model:

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

What this presentation includes:

• 2 ggplot visualizations
• 1 interactive plotly visualization
• 2 slides with LaTeX math
• 1 slide showing R code
• 1 worked example prediction

The mtcars dataset contains measurements on 32 cars from the 1970s.

Variables used here:

• mpg: miles per gallon
• wt: car weight in 1000 pounds

Practical question:

Do heavier cars tend to get worse gas mileage?

If yes, a simple linear regression model can quantify that relationship.

Simple linear regression assumes:

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\]

For this example:

\[\text{mpg}_i = \beta_0 + \beta_1 \text{wt}_i + \varepsilon_i\]

Where:

• \(\beta_0\) = intercept
• \(\beta_1\) = slope
• \(\varepsilon_i\) = random error term

The least-squares estimates choose \(b_0\) and \(b_1\) to minimize:

\[\sum_{i=1}^{n}(y_i - \hat{y}_i)^2\]

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

summary(fit)

## 
## 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

ggplot(mtcars, aes(wt, mpg)) +
  geom_point(size = 3, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "firebrick") +
  labs(
    title = "Linear Fit for mpg vs weight",
    x = "Weight (1000 lbs)",
    y = "Miles per gallon"
  ) +
  theme_minimal(base_size = 16)

From the fitted model:

\[\widehat{\text{mpg}} = 37.29 - 5.34(\text{wt})\]

Interpretation of the slope:

• For each additional 1000 pounds of weight,
• the predicted fuel efficiency changes by about -5.34 mpg.

Because the slope is negative, heavier cars are predicted to get lower gas mileage.

Worked example: for a car weighing 3,000 pounds (\(wt = 3.0\)),

\[\widehat{\text{mpg}} = 21.25\]

A good residual plot should show points scattered around 0 with no strong pattern.

To test whether weight is linearly related to mpg, we use:

\[H_0: \beta_1 = 0 \qquad \text{vs} \qquad H_a: \beta_1 \ne 0\]

The test statistic is:

\[t = \frac{b_1 - 0}{SE(b_1)}\]

From the model summary:

• Estimated slope: -5.344
• \(R^2\): 0.753
• p-value for slope: 1.29^{-10}

Since the p-value is very small, we reject \(H_0\) and conclude that weight is a significant predictor of mpg.