Simple linear regression is a statistical method used to study the relationship between two quantitative variables.

• Independent variable (x): the variable used to make a prediction
• Dependent variable (y): the variable we want to predict
• The goal is to find a straight line that best describes the relationship between x and y

For example, we can study whether a car’s weight can be used to predict its fuel efficiency (MPG).

The equation for simple linear regression is:

\[y = \beta_0 + \beta_1x + \epsilon\]

Where:

• \(y\) = dependent variable
• \(x\) = independent variable
• \(\beta_0\) = intercept
• \(\beta_1\) = slope
• \(\epsilon\) = random error

The slope tells us how much we expect y to change when x increases by one unit.

For my example, I’ll will use the built-in mtcars dataset in R.

Now let’s examine the relationship between:

• wt: car weight (in 1,000 lbs)
• mpg: fuel efficiency in miles per gallon

Our question is:

Can a car’s weight be used to predict its fuel efficiency?

We expect heavier cars to generally have lower MPG.

The scatter plot below shows the relationship between car weight and fuel efficiency.

A regression line helps us visualize the relationship between weight and fuel efficiency.

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

This interactive plot allows us to explore the relationship between weight and MPG.

For our example, the estimated regression equation has the form:

\[\widehat{MPG} = \beta_0 + \beta_1(Weight)\]

The slope \(\beta_1\) describes how MPG changes as car weight increases.

• If \(\beta_1 > 0\), MPG increases as weight increases.
• If \(\beta_1 < 0\), MPG decreases as weight increases.

=Based on our plots, we expect \(\beta_1\) to be negative because heavier cars tend to have lower MPG.

We can create a simple linear regression model using the lm() function in R.

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

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Coefficients:
## (Intercept)           wt  
##      37.285       -5.344

Our fitted regression equation is:

\[\widehat{MPG} = 37.285 - 5.344(Weight)\]

The slope is negative, which supports what we observed in our graphs.

• For every additional 1,000 lbs of weight, predicted MPG decreases by about 5.344 MPG.
• Heavier cars generally have lower fuel efficiency.
• Simple linear regression allows us to describe and predict the relationship between two quantitative variables.