These slides demonstrates simple linear regression using the built in trees data set in R.
I will analyze how a tree, its circumference in inches, relates to its volume in cubic feet.
The slides and graphs used to visualize, model, and interpret this linear relationship using ggplot2 and plotly. The model we’ll explore has the form: \[ \text{Volume}=\beta_0+\beta_1\cdot\text{Girth}+\varepsilon,\quad\mathcal{N}(0,\sigma^2) \]

My plan is to estimate \(\beta_0\) and\(\beta_1\) using least squares: \[ S(\beta_0,\beta_1)=\sum_{i=1}^n(\text{Volume}-\beta_0-\beta_1\text{Girth})^2 \]

Each observation should represents a tree, with its Girth, Height, and Volume, with a focus on Girth and Volume for estimation.

## Warning: package 'ggplot2' was built under R version 4.5.2

## Warning: package 'plotly' was built under R version 4.5.2

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

##   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

##      Girth           Height       Volume     
##  Min.   : 8.30   Min.   :63   Min.   :10.20  
##  1st Qu.:11.05   1st Qu.:72   1st Qu.:19.40  
##  Median :12.90   Median :76   Median :24.20  
##  Mean   :13.25   Mean   :76   Mean   :30.17  
##  3rd Qu.:15.25   3rd Qu.:80   3rd Qu.:37.30  
##  Max.   :20.60   Max.   :87   Max.   :77.00

The plot shows a strong upward trend the larger girths generally correspond to larger tree volumes

The points are randomly scattered around zero, suggesting a good linear fit.

Hover over any point to see its exact values an easier way to explore the mode and data.

These coefficients define the fitted regression line

The coefficients define the fitted regression line.
i WILL estimate the intercept \(\beta_0\) and slope \(\beta_1\) of the model

\[ \widehat{y}_i=\widehat{\beta}_0+\widehat{\beta}_1 x_i \]

using the least squares formulas:

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

Both intervals are narrow, showing a good relationship in the estimated linear parameters.

Below is the R code I used to generate the 90% prediction interval plot shown on the next slide(formatted to fit)

new<- data.frame(Girth=c(8, 12, 16, 20))
pred<- predict(mod, newdata=new, interval="prediction",level=0.9)
pred_tbl<- cbind(new, round(pred, 2))
#kable(pred_tbl, caption = paste0("Predicted Tree Volumes"))
ggplot(pred_tbl,aes(x= Girth, y = fit))+
  geom_ribbon(aes(ymin = lwr,ymax = upr),
              alpha = 0.2, fill= "darkgreen")+
  geom_line(color = "darkgreen")+geom_point(size =2,color= "brown")+
  labs(title="Predicted Volume vs.Girth(90%)",x="Girth (inches)",
       y="Predicted Volume (ft³)")+
  theme_minimal(base_family = "serif")

Predictions show how estimated volume grows as girth increases, with wider intervals for larger trees.