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

In this presentation, we use the mtcars dataset and study:

• wt = weight of car
• mpg = miles per gallon

We want to understand how the weight of a car affects its fuel efficiency.

Simple linear regression helps us:

• understand the relationship between two variables
• predict the value of one variable using another
• measure whether the relationship is positive or negative

Example question:

How does car weight affect fuel efficiency?

The simple linear regression model is:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

Where:

• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\varepsilon\) is the random error term

For our example:

\[ mpg = \beta_0 + \beta_1(wt) + \varepsilon \]

The estimated regression line is:

\[ \hat{y} = b_0 + b_1x \]

The slope tells us how much the predicted value of \(y\) changes when \(x\) increases by 1 unit.

\[ b_1 = \frac{\text{change in predicted } y}{\text{change in } x} \]

If \(b_1 < 0\), then as \(x\) increases, \(y\) tends to decrease.

In this example, heavier cars tend to have lower miles per gallon.

From the fitted model:

• Intercept = 37.285
• Slope = -5.344
• \(R^2 = 0.753\)

Interpretation:

• When car weight increases by 1 unit, predicted miles per gallon decreases by about 5.344.
• About 75.3% of the variation in mpg is explained by wt.

Suppose a car weighs 3.0 thousand pounds.

The fitted model predicts fuel efficiency of about 21.25 miles per gallon.

## R Code Used for a Plot


``` r
ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(size = 2.5) +
  geom_smooth(method = "lm", se = TRUE) +
  labs(
    title = "Car Weight vs Miles per Gallon",
    x = "Weight (1000 lbs)",
    y = "Miles per Gallon"
  )

Simple linear regression is a useful statistical tool for studying the relationship between two variables.

From this example, we observe that:

• heavier cars tend to have lower fuel efficiency
• regression summarizes the relationship clearly
• graphs help visualize the trend
• prediction is possible after fitting the model

Simple linear regression is widely used in science, engineering, business, and data analysis.