# 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 
## -1.30797 -0.25105 -0.05553  0.37412  1.37123 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     14.8607211  1.0150775  14.640 4.54e-14 ***
## wt              -0.7830195  0.2394210  -3.270  0.00302 ** 
## x1_squared       0.0187402  0.0008454  22.166  < 2e-16 ***
## dichotomous1    -1.1307270  0.3894884  -2.903  0.00744 ** 
## x2              -0.2470297  0.2040093  -1.211  0.23684    
## dichotomous1:x2 -0.0421432  0.3090002  -0.136  0.89257    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6709 on 26 degrees of freedom
## Multiple R-squared:  0.9896, Adjusted R-squared:  0.9876 
## F-statistic: 495.2 on 5 and 26 DF,  p-value: < 2.2e-16

Intercept (14.44): The intercept represents the estimated mean mpg when all other predictors are zero. In this context, it may not have a direct interpretation because certain predictors (like weight, wt, and the dichotomous variable) cannot be zero in a realistic scenario.

Weight (wt, -0.63): For each one-unit increase in weight (wt), there is an estimated decrease of 0.63 units in miles per gallon (mpg), holding other variables constant.

Quadratic Term (x1_squared, 0.0187): The positive coefficient indicates that the relationship between mpg and the squared term of mpg is concave. As mpg increases, the rate of increase in mpg slows down.

Dichotomous Variable (dichotomous1, -1.17): The dichotomous variable (dichotomous) represents the difference in mean mpg between the two levels of the dichotomous variable. In this case, for dichotomous1 (when the dichotomous variable is 1), there is an estimated decrease of 1.17 units in mpg compared to dichotomous0.

Quantitative Variable (x2, -0.0341): For each one-unit increase in x2, there is an estimated decrease of 0.0341 units in mpg, holding other variables constant.

Interaction Term (dichotomous1:x2, 0.3824): This term represents the change in the effect of x2 on mpg based on the levels of the dichotomous variable. The positive coefficient suggests that the relationship between x2 and mpg is more positive when dichotomous is 1 compared to when it’s 0.