Discussion Response

A multiple linear regression model is a generalization of the simple linear regression model It has \(k\) variables with the form: \[ \hat{y}=a_0+a_1 x_1+a_2 x_2+\cdots a_k x_k, \] where the \(x_i\) values are the inputs to the system, the \(a_i\) coefficients are the model parameters computed from the measured data, and \(\hat{y}\) is the output value predicted by the model.

# Load the mtcars dataset
data(mtcars)

# Add the required terms to the dataset
mtcars$x1_squared <- mtcars$mpg^2  # Quadratic term
mtcars$dichotomous <- as.factor(ifelse(mtcars$cyl > median(mtcars$cyl), 1, 0))  # Dichotomous term
mtcars$x2 <- rnorm(nrow(mtcars))  # Quantitative variable for interaction

# Build a multiple regression model
model <- lm(mpg ~ wt + x1_squared + dichotomous + x2 + dichotomous:x2, data = mtcars)

# Display the model summary
summary(model)
## 
## Call:
## lm(formula = mpg ~ wt + x1_squared + dichotomous + x2 + dichotomous:x2, 
##     data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.3967 -0.3000 -0.1028  0.3119  1.4706 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     14.9279364  1.0479404  14.245 8.58e-14 ***
## wt              -0.7708947  0.2442802  -3.156  0.00402 ** 
## x1_squared       0.0183835  0.0008503  21.619  < 2e-16 ***
## dichotomous1    -1.0192270  0.3720813  -2.739  0.01097 *  
## x2               0.1286721  0.1796677   0.716  0.48027    
## dichotomous1:x2 -0.3536109  0.2638667  -1.340  0.19180    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.691 on 26 degrees of freedom
## Multiple R-squared:  0.989,  Adjusted R-squared:  0.9869 
## F-statistic: 466.4 on 5 and 26 DF,  p-value: < 2.2e-16
# Extract residuals from the transformed linear regression model
residual <- residuals(model)

# Create a histogram of residuals
hist(residual, breaks = 20, col = "skyblue", border = "black", 
     main = "Histogram of mtcars Residuals", xlab = "Residuals", ylab = "Frequency")

The histogram seems to be bell shaped and looks nearly residuals are normal distributed. It might be said that the multiple linear regression model fit with the data set.