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

We use one variable, \(x\), to predict another variable, \(y\).

Example: predicting stopping distance from speed using the built-in cars dataset in R.

Simple linear regression helps us: - understand trends in data - measure how strongly two variables are related - make predictions

It is commonly used in business, biology, engineering, and data science.

The simple linear regression model is:

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

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

The regression line is chosen to minimize the sum of squared residuals:

\[ \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 \]

This is called the least squares criterion.

Residuals are the vertical distances between observed values and predicted values.

We will use the built-in cars dataset.

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

The variables are: - speed: speed of a car - dist: stopping distance

We want to see whether stopping distance tends to increase as speed increases.

This plot shows a positive relationship between speed and stopping distance.

As speed increases, stopping distance generally increases as well.

This residual plot helps us check whether the linear model is reasonable.

If the residuals are randomly scattered around zero, the linear model is more appropriate.

This interactive plot lets the viewer hover over points and inspect the data more closely.

Interactive visualizations make it easier to explore individual observations.

The slope tells us how much stopping distance is expected to change for a one-unit increase in speed.

A positive slope suggests that faster cars generally need more distance to stop.

model <- lm(dist ~ speed, data = cars)

ggplot(cars, aes(x = speed, y = dist)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(title = "Regression Line for Cars Data")

Simple linear regression is a foundational statistical tool for modeling relationships between variables.

In this example, we saw that stopping distance tends to increase as speed increases.

Using ggplot and plotly makes it easier to communicate both static and interactive insights from data.