rm(list = ls()) #clear environment and remove all files from the workspace

gc()   #clear the unused memory
##          used (Mb) gc trigger (Mb) limit (Mb) max used (Mb)
## Ncells 525900 28.1    1167871 62.4         NA   669400 35.8
## Vcells 968912  7.4    8388608 64.0      16384  1851644 14.2

Choosing Dataset

df = sleep
head(df)
##   extra group ID
## 1   0.7     1  1
## 2  -1.6     1  2
## 3  -0.2     1  3
## 4  -1.2     1  4
## 5  -0.1     1  5
## 6   3.4     1  6

About the data:
extra : increase in hours of sleep

group : drug given

ID : patient ID

summary(df)
##      extra        group        ID   
##  Min.   :-1.600   1:10   1      :2  
##  1st Qu.:-0.025   2:10   2      :2  
##  Median : 0.950          3      :2  
##  Mean   : 1.540          4      :2  
##  3rd Qu.: 3.400          5      :2  
##  Max.   : 5.500          6      :2  
##                          (Other):8

A.

The dependent variable would by extra (Y) and the independent variable would be drug given(X).

B.

# Creating model
model <- lm(extra ~ group, data = df)

# Print the model summary
print(model)
## 
## Call:
## lm(formula = extra ~ group, data = df)
## 
## Coefficients:
## (Intercept)       group2  
##        0.75         1.58
# Scatter plot with regression line for each level of 'group'
plot(extra ~ group, 
     data = df, 
     main = "Increase in Hours of Sleep vs. Drug Given", 
     xlab = "Drug Given", ylab = "Increase in Hours of Sleep")

# Adding regression lines for each level of 'group'
abline(lm(extra ~ group, data = df), col = "blue")

C.

When group = 1, the extra hours of sleep is 0.75 while when the group is 2, the hours of sleep increarses by 1.58.

D.

Slope: \[\beta_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}\]

X <- as.numeric(df$group)
Y <- df$extra
covariance <- cov(X, Y)
variance <- var(X)

slope <- covariance / variance
slope
## [1] 1.58

Intercept:

\[\beta_0 = \bar{Y} - \beta_1 \bar{X}\]

# Calculate mean of X and Y
mean_X <- mean(X)
mean_Y <- mean(Y)

intercept <- mean_Y - (slope * mean_X)
intercept
## [1] -0.83