Simple linear regression studies the relationship between two variables using a straight line.

• The predictor variable (x) is used to predict another variable.
• The response variable (y) is what we want to predict.

The fitted regression line has the equation:

\[ \hat{y} = a + bx \]

• \(\hat{y}\) is the predicted response.
• \(a\) is the y-intercept.
• \(b\) is the slope.
• \(x\) is the predictor.

We will use the built in ‘iris’ dataset.

We want to see whether petal length can help predict petal width.

• Predictor (x): ‘Petal.Length’
• Response (y): ‘Petal.Width’

The colors represent the three iris species.

We use ‘lm()’ to predict petal width from petal length.

mod = lm(Petal.Width ~ Petal.Length, data = iris)
mod

## 
## Call:
## lm(formula = Petal.Width ~ Petal.Length, data = iris)
## 
## Coefficients:
##  (Intercept)  Petal.Length  
##      -0.3631        0.4158

The fitted regression equation is:

\[ \hat{y} = -0.3631 + 0.4158x \]

Here, \(x\) is petal length and \(\hat{y}\) is predicted petal width.

For each 1 cm increase in petal length, predicted petal width increases by approximately 0.4158 cm.

The positive slope indicates a positive relationship between petal length and petal width.

Petal width generally increases as petal length increases.

• Petal length and petal width have a positive relationship.
• Simple linear regression can help predict petal width from petal length.