• Simple linear regression looks at how one number affects another.
• It fits a straight line through the data.
• Analysts use it to predict things like sales, cost, or demand.
• We will build one step by step using R.

A straight line through the data looks like this:

\[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i \]

• \(\beta_0\) is the intercept, where the line starts.
• \(\beta_1\) is the slope, how steep the line is.
• \(\varepsilon_i\) is the error, the part the line does not explain.

R finds the best line using least squares:

\[ \hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}, \qquad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x} \]

• Least squares picks the line that keeps the errors as small as possible.
• In R, one function does all of this: lm()

Using the trees dataset built into R. It has the girth, height, and volume data of trees.

head(trees)

##   Girth Height Volume
## 1   8.3     70   10.3
## 2   8.6     65   10.3
## 3   8.8     63   10.2
## 4  10.5     72   16.4
## 5  10.7     81   18.8
## 6  10.8     83   19.7

We will predict Volume from Girth.

• Simple linear regression fits a straight line through two variables.
• The slope and intercept come from least squares, found in R with lm().
• ggplot2 and plotly can both show the fit and plotly adds zoom and hover.
• With more than one predictor, a 3D plot can show everything at once.