1. Understand the linear regression model for tree volume.
2. Explore relationships between Volume, Girth, and Height.
3. Fit linear models and interpret slope estimates.
4. Display results using scatterplots and 3D interactive plots.
5. Display an example of R code for a 3D interactive plot.

To model tree volume (\(Y = \text{Volume}\)) using girth (\(x = \text{Girth}\)), we have:

\[ Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad \varepsilon_i \sim N(0,\sigma^2) \] \(\beta_1\) is the expected change in volume as girth increases for each inch.

Least squares estimates minimize \(\sum_i (Y_i - \beta_0 - \beta_1 x_i)^2\).

Hypothesis test for slope: \[ H_0: \beta_1 = 0 \quad \text{vs}\quad H_a: \beta_1 \neq 0 \] t-statistic: \[ t = \frac{\hat\beta_1 - 0}{\text{SE}(\hat\beta_1)}, \quad \text{df} = n-2 \]

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

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

# 3D interactive scatterplot of Volume, Girth, and Height

library(plotly)
plot_ly(
  data = trees, 
  x = ~Girth,
  y = ~Height, 
  z = ~Volume,
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 6, color = ~Volume, colorscale = "Viridis")
)