It is the first step on learning Statistics!

Finding out the relationship between two measures. For example: weight and height?, sleep and mood?.

Draw the best straight line through the points. (this line shows the pattern).

Use that line to make predictions. If I know enough of a topic I can make better predictions on it.

The model is:

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

Where:

• \(y\) = the value we want to predict
  
• \(x\) = the value we use to predict
  
• \(\beta_0\) = intercept (baseline when x = 0)
  
• \(\beta_1\) = slope (how much y changes when x increases by 1)
  
• \(\varepsilon\) = error (the randomness/noise)

We use simple linear regression to:

• See if two things are related
  (Do they move together?)

• Understand the direction of the relationship

  • If one goes up, does the other go up or down?

• Measure how strong the relationship is

• Draw a line that shows the pattern

• Make predictions
  If we know one value (X), we can guess the other (Y)

• More hours studied → higher test score?
  
• More weight → lower gas mileage?
  
• More sleep → better mood?

We use the least squares method to find the line.

The slope of the line is:

\[ \beta_1 = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2} \]

The intercept of the line is:

\[ \beta_0 = \bar{y} - \beta_1 \bar{x} \]

These formulas give us the best-fitting line that minimizes the distance between the points and the line.

We will use the built-in pressure dataset in R.

It contains measurements of:

• temperature (in degrees Celsius)
• pressure (in mm of mercury)

We want to see how temperature affects pressure.

As temperature increases, pressure also increases.

This makes it a good example for simple linear regression.

Before building the model, let’s look at the first few rows of the pressure dataset.

library(ggplot2)

ggplot(pressure, aes(x = temperature, y = pressure)) +
  geom_point(size = 3) +
  labs(title = "Scatterplot of Temperature vs Pressure",
       x = "Temperature (°C)",
       y = "Pressure (mm of mercury)")

Now let’s take a closer look at the scatterplot so we can really see its point.

library(ggplot2)

ggplot(pressure, aes(x = temperature, y = pressure)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = FALSE, color = "blue") +
  labs(title = "Temperature vs Pressure with Regression Line",
       x = "Temperature (°C)",
       y = "Pressure (mm of mercury)")

The line shows the general trend, but it doesn’t fit the data very well because the relationship is curved, not perfectly straight.

• Purpose: To Model and predict a response \(y\) from one predictor \(x\) with a straight line.
  
• What it gives: A clear trend direction (slope), an intercept, and simple predictions.
  
• When it works well: The relationship between \(x\) and \(y\) is approximately linear, with constant variance and independent errors.
• Limits: If the true pattern is curved or shows changing spread, a straight line can mislead or fit poorly.