• What linear regression is
  
• The mathematical model
  
• An example using real data
  
• Diagnostics and prediction
  
• Visualization with ggplot and Plotly

Linear regression is used to explain and predict a numeric outcome using one or more predictors.

Examples: - Business: sales vs advertising
- Engineering: output vs system inputs
- Biology: growth vs dosage

In this presentation: - Response variable: mpg (miles per gallon)
- Predictor: wt (car weight)

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

Where: - \(\beta_0\) is the intercept
- \(\beta_1\) is the slope
- \(\varepsilon_i\) is random error

\[ \min_{\beta_0,\beta_1}\sum_{i=1}^{n}(Y_i-\beta_0-\beta_1X_i)^2 \]

This gives the “best-fitting” line.

Interpretation: - Negative slope → heavier cars have lower mpg
- p-value tests whether weight is a significant predictor

A good model shows residuals randomly scattered around zero.

For a new value \(x_0\):

\[ \hat{Y}(x_0)=\hat{\beta}_0+\hat{\beta}_1 x_0 \]

Two intervals: - Confidence interval: mean response
- Prediction interval: new observation (wider)

# Fit the model
fit1 <- lm(mpg ~ wt, data = mtcars)

# Predict mpg for a car weighing 3.0 (1000 lbs)
predict(fit1, newdata = data.frame(wt = 3.0), interval = "prediction")

##        fit      lwr      upr
## 1 21.25171 14.92987 27.57355

• Linear regression models relationships between variables
  
• Least squares finds the best-fitting line or plane
  
• ggplot and Plotly improve interpretation and communication