Simple linear regression models the relationship between two variables by fitting a linear equation to observed data.
We use it to predict outcomes and understand associations. In this project, we apply it to real-world car data to see how weight impacts fuel efficiency.

• Understand the simple linear regression model and OLS estimation
• Fit and interpret a regression line in R
• Produce ggplot diagnostics and interactive Plotly visuals(2D and 3D)
• Show the R code used to generate the results (These objectives guide us through both theory and application of regression)

The simple linear regression model:
\[ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, \qquad \epsilon_i \sim iid(0, \sigma^2) \]

Where:
- \(Y_i\) = dependent variable
- \(X_i\) = independent variable
- \(\beta_0\) = intercept
- \(\beta_1\) = slope
- \(\epsilon_i\) = error term

This equation describes how we estimate the outcome(Y) using predictor(X).

Ordinary least squares (OLS) estimates:
\[ \hat{\beta}_1 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{n} (X_i - \bar{X})^2}, \qquad \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} \] Variance of the slope estimator:

\[ Var(\hat{\beta}_1) = \frac{\sigma^2}{\sum_{i=1}^{n} (X_i - \bar{X})^2} \] These formulas show us how regression finds the best fitting line by minimizing squared errors.

For this project, we will use the “mtcars” dataset to evaluate the relationship between car weight and fuel efficiency.
Simple linear regression is commonly used to quantify relationships between a predictor and outcome, so weight will be our independent variable and miles per gallon will be our dependent variable.
We also add a “car” column to label points in interactive plots.

df <- mtcars[, c("mpg", "wt", "hp")]
df$car <- rownames(mtcars)
head(df)

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

This code produces a scatterplot with the regression line to visualize the model fit.

ggplot(df, aes(x = wt, y = mpg)) +
  geom_point(size = 2) +
  geom_smooth(method = "lm", formula = y ~ x, se = TRUE, color = "blue") +
  labs(title = "MPG vs Weight with OLS Line",
       x = "Weight (1000 lbs)", y = "MPG") +
  theme_minimal(base_size = 14)

Residual plots help check assumptions like linearity and constant variance.

fitted_vals <- fitted(fit)
residuals_vals <- resid(fit)

plot(fitted_vals, residuals_vals,
     main = "Residuals vs Fitted",
     xlab = "Fitted values", ylab = "Residuals",
     pch = 19)
abline(h = 0, lty = 2, col = "red")

Shows us whether residuals are approximately normally distributed.

hist(residuals(fit), breaks=10, col='steelblue', border='black',
     main='Histogram of Residuals', xlab='Residual', ylab='Count')

Q-Q plot confirms whether residuals follow a normal distribution assumption

qqnorm(residuals(fit), main='Q-Q Plot of Residuals', cex=0.6)
qqline(residuals(fit), col='red')

This allows us to hover over points and see individual car details.

p_plotly <- plot_ly(df, x = ~wt, y = ~mpg, type = 'scatter', mode = 'markers',
                    text = ~paste("car:", car, "<br>wt:", wt, "<br>mpg:", mpg),
                    hoverinfo = 'text', marker = list(size=5)) %>%
  add_lines(x = ~wt, y = fitted(fit), line = list(color = 'blue')) %>%
  layout(height=300)
p_plotly

Here, we extend the model to multiple regression with two predictors: weight and horsepower.

# 3D regression plane code (shown as code only)
fit3 <- lm(mpg ~ wt + hp, data = df)

wt_seq <- seq(min(df$wt), max(df$wt), length.out = 30)
hp_seq <- seq(min(df$hp), max(df$hp), length.out = 30)
grid <- expand.grid(wt = wt_seq, hp = hp_seq)
grid$pred <- predict(fit3, newdata = grid)
zmat <- matrix(grid$pred, nrow = 30, ncol = 30)

The 3D surface shows how weight and horsepower together affect fuel efficiency.

Test significance of slope:

\[ H_0: \beta_1 = 0 \quad \text{vs.} \quad H_A: \beta_1 \neq 0 \]

Test statistic:

\[ t = \frac{\hat{\beta}_1 - 0}{SE(\hat{\beta}_1)} \]

Compare with critical \(t\)-value or p-value. This tests whether car weight has a significant effect on MPG.

We use a model to predict MPG for cars of different weights and show both confidence and prediction intervals.

newdata <- data.frame(wt = c(2.5, 3.0, 3.5))
conf_int <- predict(fit, newdata = newdata, interval = "confidence")
pred_int <- predict(fit, newdata = newdata, interval = "prediction")
list(confidence = conf_int, prediction = pred_int)

## $confidence
##        fit      lwr      upr
## 1 23.92395 22.55284 25.29506
## 2 21.25171 20.12444 22.37899
## 3 18.57948 17.43342 19.72553
## 
## $prediction
##        fit      lwr      upr
## 1 23.92395 17.55411 30.29378
## 2 21.25171 14.92987 27.57355
## 3 18.57948 12.25426 24.90469

• Always check linearity with scatterplots
  
• Check homoscedasticity (residuals vs fitted)
  
• Check residual normality (histogram & Q-Q plot)
  
• Use ggplot and plotly to visualize data and regression
  
• Interactive plots allow inspecting individual observations