Problem: A pizza shop wants to estimate delivery times for their customers.
Simple idea: The farther away the customer lives, the longer pizza delivery takes.
Goal: To build a model to predict delivery time estimates based on the distance.
We want to draw the best straight line through our data.
The equation: \[Y = \beta_0 + \beta_1 X\]
Example: If \(\beta_0 = 15\) and \(\beta_1 = 3\), then a 5-mile delivery takes: \(15 + 3(5) = 30\) minutes
| Distance (miles) | Time (minutes) |
|---|---|
| 3.2 | 20.9 |
| 8.0 | 31.4 |
| 4.4 | 24.8 |
| 8.9 | 36.0 |
| 9.4 | 37.2 |
| 0.9 | 15.7 |
| 5.5 | 26.9 |
| 9.0 | 34.3 |
| 5.7 | 25.7 |
| 4.8 | 23.3 |
60 deliveries recorded over one week
Clear positive relationship!
Our fitted equation: \[\widehat{\text{Time}} = 12.51 + 2.44 \times \text{Distance}\]
What this derives:
Example prediction:
For a 6-mile delivery: \(12.51 + 2.44 \times 6 = 27.2\) minutes
# Basic regression in R - it's this simple! model <- lm(Time ~ Distance, data = pizza_data) summary(model) predict(model, newdata = data.frame(Distance = 6))
That’s it! Just one line to fit the model.
| Metric | Value | |
|---|---|---|
| R-squared | 0.94 | |
| (Intercept) | Base Time (minutes) | 12.51 |
| Distance | Time per Mile (minutes) | 2.44 |
R² = 0.94 means distance explains 94% of delivery time variation.
Pretty good!
| Distance | Predicted Time | Lower | Upper |
|---|---|---|---|
| 2 | 17.4 | 13.9 | 20.9 |
| 5 | 24.7 | 21.3 | 28.2 |
| 8 | 32.0 | 28.6 | 35.5 |
Example: 5-mile delivery takes about 25 minutes (give or take 3 minutes)
Use: Pizza shop can now give accurate time estimates to all the customers!
Key takeaway: Linear regression helps us make predictions from simple relationships and data.