Simple linear regression is a statistical method used to model the relationship between one predictor variable and one response variable.

In this presentation, we use car weight to predict fuel efficiency.

\[ Y = \beta_0 + \beta_1 X + \epsilon \]

where:

• \(Y\) is the response variable
  
• \(X\) is the predictor variable
  
• \(\beta_0\) is the intercept
  
• \(\beta_1\) is the slope
  
• \(\epsilon\) is the random error

\[ \hat{Y} = b_0 + b_1 X \]

The coefficients \(b_0\) and \(b_1\) are chosen using the least squares method, which minimizes the sum of squared residuals.

We use the built-in R dataset mtcars.

Variables used:

• mpg = miles per gallon
  
• wt = weight of the car
  
• hp = horsepower

model <- lm(mpg ~ wt, data = mtcars)
summary(model)

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

Simple linear regression helps explain how one variable changes as another variable changes.

It is widely used in statistics, data science, economics, and engineering.