• Quick intro to Simple Linear Regression
• Data example: car weight vs. fuel efficiency
• Fit the model, test the slope, check assumptions
• Plots and key takeaways

• Predict one numeric value from another.
• Examples:
  • Fuel efficiency vs. car weight
  • Blood pressure vs. sodium intake
• We’ll use the built-in mtcars dataset.

We model \[ Y = \beta_0 + \beta_1 X + \varepsilon \] where: - \(Y\): response (mpg) - \(X\): predictor (weight) - \(\varepsilon\): random error

1. Linearity of mean response
2. Constant variance of errors
3. Independence of observations
4. Errors roughly normal for inference

library(tidyverse)
library(broom)
library(plotly)

data(mtcars)
df <- mtcars %>% select(mpg, wt, hp)
head(df)

##                    mpg    wt  hp
## Mazda RX4         21.0 2.620 110
## Mazda RX4 Wag     21.0 2.875 110
## Datsun 710        22.8 2.320  93
## Hornet 4 Drive    21.4 3.215 110
## Hornet Sportabout 18.7 3.440 175
## Valiant           18.1 3.460 105

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

ggplot(df, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = TRUE) +
  labs(title="Fuel Efficiency vs Weight",
       x="Weight (1000 lbs)", y="MPG")

• Slope \(\hat{\beta}_1\): change in mpg for each 1000 lb increase in weight.
• Example output:

##      Estimate    Std. Error       t value      Pr(>|t|) 
## -5.344472e+00  5.591010e-01 -9.559044e+00  1.293959e-10

Test if slope differs from 0: \[ H_0: \beta_1 = 0, \quad H_a: \beta_1 \neq 0 \] t-statistic: \[ t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)} \]

aug <- augment(fit)
ggplot(aug, aes(.fitted, .resid)) +
  geom_hline(yintercept=0, linetype="dashed") +
  geom_point() +
  labs(title="Residuals vs Fitted", x="Fitted", y="Residuals")

plot_ly(df, x=~wt, y=~hp, z=~mpg,
        type="scatter3d", mode="markers") %>%
  layout(scene=list(
    xaxis=list(title="Weight"),
    yaxis=list(title="Horsepower"),
    zaxis=list(title="MPG")
  ))

• SLR predicts a numeric outcome from one predictor.
• Check assumptions with residual plots.
• In our example: heavier cars → lower mpg.
• Plotly adds interactive 3D insights.