• Simple linear regression is a statistical technique used to model the relationship between a dependent variable and an independent variable.
• In this example, we will use the mtcars dataset to build a simple linear regression model to predict the miles per gallon (mpg) of a car based on its horsepower.
• This means that horsepower is is the independent variable X and miles per gallon is the dependent variable Y.

• The mtcars data set contains information about different car models, including the number of cylinders, horsepower, miles per gallon, and other variables.
• The data set has 32 observations and 11 variables, but we will focus on only two variables, hp and mpg.

• We can use this piece of code to do this.

library(ggplot2)
ggplot(mtcars, aes(x=hp, y=mpg)) +
  geom_point() +
  xlab("Horsepower") +
  ylab("Miles per Gallon") +
  ggtitle("Scatter Plot of Horsepower and Miles per Gallon")

• We can see that there is a negative relationship between horsepower and miles per gallon just by looking at the points but we can show this mathematically using a regression line.

• The linear regression model can be expressed as: \[ Y = \beta_0 + \beta_1 X + \epsilon \]
• where Y is the dependent variable, X is the independent variable, \(\beta_0\) is the intercept, \(\beta_1\) is the slope, and \(\epsilon\) is the error term.
• We can use the lm() function in R to fit the regression line.

  	x
  (Intercept)	30.0988605
  hp	-0.0682283

• so here we would have \(\beta_0 = 30.0988605\) and \(\beta_1 = -0.0682283\)

• The regression line supports our earlier conclusion that there is a negative relation ship between the amount of horsepower a car has and the miles per gallon its engine gets.

## `geom_smooth()` using formula = 'y ~ x'

• The summary() function provides an overview of the model’s performance

## 
## Call:
## lm(formula = mpg ~ hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7121 -2.1122 -0.8854  1.5819  8.2360 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 30.09886    1.63392  18.421  < 2e-16 ***
## hp          -0.06823    0.01012  -6.742 1.79e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared:  0.6024, Adjusted R-squared:  0.5892 
## F-statistic: 45.46 on 1 and 30 DF,  p-value: 1.788e-07

• The residual plot is a useful tool to check the linear regression model.
• The residuals are the difference between the actual and predicted values of the dependent variable.

• we can see if there is a further relationship between mpg, hp, and the cubic volume of the engine(disp) by using a 3d scatter plot.

• Simple linear regression is a useful technique for modeling the relationship between a dependent variable and an independent variable where the simple linear regression creates a model that minimizes the error given by the square of the residuals.
• The regression line given by \(Y = \beta_0 + \beta_1 X + \epsilon\) can be used to predict the dependent variable based on the independent variable with a given error \(\epsilon\)
• This allows for the modeling of simple relationships within our data to help us understand what trends we might be able to dive further into with more advanced techniques.