📘 11.1 – Simple Linear Regression Summary

📈 Scatter Plot – Observing the Pattern

Let's start with a scatter plot of real data. Below is the relationship between hydrocarbon level (x) and oxygen purity (y):

From the plot, we can tell that there's a general pattern: as the hydrocarbon level increases, oxygen purity tends to increase. This suggests a potential linear relationship, even though the data does not fall perfectly on a straight line.

📌 Regression Model

Y = β₀ + β₁x + ε E(Y|x) = β₀ + β₁x Var(Y|x) = σ²

Y: Response variable (e.g., oxygen purity)
x: Predictor variable (e.g., hydrocarbon level)
β₀: Intercept = 75
β₁: Slope = 15
ε: Error term, mean 0, variance σ² = 2

📘 Deriving E(Y|x) and Var(Y|x)

Let the model be: Y = β₀ + β₁x + ε Expected value: E(Y|x) = E(β₀ + β₁x + ε) = β₀ + β₁x + E(ε) = β₀ + β₁x (Assuming E(ε) = 0) Variance: Var(Y|x) = Var(β₀ + β₁x + ε) = Var(β₀ + β₁x) + Var(ε) = 0 + σ² = σ² (β₀ + β₁x is constant, so its variance is 0)

🧪 Model Example

μ Y|x = 75 + 15x, σ² = 2

Hydrocarbon Level (x): 1.23

Predicted Oxygen Purity (μY): 93.45%

⚠️ Warnings

Correlation ≠ Causation: Just because two variables move together doesn’t mean one causes the other.
Extrapolation Risk: Predicting far outside the range of x is unreliable.

🧠 Key Vocabulary (with Examples)

Term	Definition	Example
Scatterplot	A graph that displays paired data points (x, Y).	Plotting hydrocarbon level (x) vs. oxygen purity (Y) to observe a pattern.
Regression	Describes how one variable (Y) depends on another (x).	Modeling oxygen purity based on hydrocarbon level.
Simple Linear Regression	A model with one independent variable (x).	`Y = β₀ + β₁x + ε`
Slope (β₁)	The change in Y for each unit increase in x.	If `β₁ = 15`, then each 1% increase in x raises Y by 15%.
Intercept (β₀)	The predicted value of Y when x = 0.	If `β₀ = 75`, then Y is 75 when x = 0.
Error Term (ε)	Represents variation in Y not explained by x.	Noise from temperature or equipment that affects oxygen purity.