Clear Data:

rm(list = ls())      # Clear all files from your environment
         gc()            # Clear unused memory
##          used (Mb) gc trigger (Mb) limit (Mb) max used (Mb)
## Ncells 526513 28.2    1169612 62.5         NA   669420 35.8
## Vcells 970825  7.5    8388608 64.0      16384  1851931 14.2
         cat("\f")       # Clear the console
 graphics.off()      # Clear all graphs

Familiarizing with data:

head(Orange)
##   Tree  age circumference
## 1    1  118            30
## 2    1  484            58
## 3    1  664            87
## 4    1 1004           115
## 5    1 1231           120
## 6    1 1372           142
summary(Orange)
##  Tree       age         circumference  
##  3:7   Min.   : 118.0   Min.   : 30.0  
##  1:7   1st Qu.: 484.0   1st Qu.: 65.5  
##  5:7   Median :1004.0   Median :115.0  
##  2:7   Mean   : 922.1   Mean   :115.9  
##  4:7   3rd Qu.:1372.0   3rd Qu.:161.5  
##        Max.   :1582.0   Max.   :214.0

Part A

  1. Dependent Variable (Y):

    • Circumference (mm): This is the dependent variable. It represents the measurement of the circumference of the trunk circumference of the orange tree and is affected by the independent variable, age.

  2. Independent Variable (X):

    • Age (days): This variable represents the age of the orange tree in days. It is considered the independent variable, influencing or explaining changes in the circumference of the orange tree.

\(Y_i = \beta_0 + \beta_1 * X_i + \epsilon_i\)

\(Y_i\) = circumference of the \(i\)th orange tree

\(X_i\) = Age of the \(i\)th orange tree

\(\beta_0\) = Interecept (constant) term

\(\beta_i\) = Slope coefficient representing the change in circumference (mm) per unit change in age

\(\epsilon_i\) = Error term representing enexplainted variation

library(ggplot2)

# assign data set
data <- Orange

# Correlation
cor(data$circumference, 
    data$age, 
    use = "complete.obs")
## [1] 0.9135189
# Graph
ggplot(data, aes(x = age, 
                 y = circumference)) +
  geom_point(size = 2, 
             shape = 18, 
             col = "red") +
  stat_smooth(method = lm, 
              linetype = "dashed") + 
  xlab("Age (days)") + 
  ylab("Circumference (mm)")
## `geom_smooth()` using formula = 'y ~ x'

Part B

lm_model <- lm(circumference ~ age, 
               data = data)

# Summary of the linear regression model
summary(lm_model)
## 
## Call:
## lm(formula = circumference ~ age, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -46.310 -14.946  -0.076  19.697  45.111 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 17.399650   8.622660   2.018   0.0518 .  
## age          0.106770   0.008277  12.900 1.93e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.74 on 33 degrees of freedom
## Multiple R-squared:  0.8345, Adjusted R-squared:  0.8295 
## F-statistic: 166.4 on 1 and 33 DF,  p-value: 1.931e-14

Part C

Intercept \(\beta_0\) = Estimated circumference of the orange tree trunk at 0 days, this might not be applicable in this scenario though as the orange tree wouldn’t have a circumference at 0 days old.

Slope \(\beta_i\) = The change in circumference for each day in age, in this case the coefficient is 0.107, meaning that each day of age the circumference might increase an estimated 0.10mm

Part D

Slope

\(\beta_i\) = \(\frac{cov(X,Y)}{Var(X)}\)

\(cov(X,Y)\) = Covariance between X (age) and Y (circumference)

\(Var(X)\) = Variance X (age)

Covariance <- cov(data$age,data$circumference)
print(Covariance)
## [1] 25831.02
Variance <- var(data$age)
print(Variance)
## [1] 241930.7
Bi <- Covariance/Variance
print(Bi)
## [1] 0.1067703

The 0.1068 matches the coefficient we got in Part B

Intercept

\(\beta_0\) = \(Y - \beta_i * X\)

\(Y\) = Mean circumference

\(X\) = Mean age

Y <- mean(data$circumference)
print(Y)
## [1] 115.8571
X <- mean(data$age)
print(X)
## [1] 922.1429
B0 <- Y - Bi * X
print(B0)
## [1] 17.39965

The 17.40 matches the coefficient we got in Part B