Set seed for reproducibility

set.seed(123)

1. Simulate 2000 values of X

n <- 2000 x <- rnorm(n) # Now x is defined!

2. Define your model’s coefficients from part (d)

beta0 <- -1.8 beta1 <- -2.6

3. Compute your model’s probability of “orange”

prob_your <- 1 / (1 + exp(-(beta0 + beta1 * x)))

4. Generate class predictions based on your model

y_your_pred <- ifelse(prob_your > 0.5, “orange”, “apple”)

5. Friend’s model coefficients (softmax form)

alpha_orange0 <- 1.2 alpha_orange1 <- -2 alpha_apple0 <- 3 alpha_apple1 <- 0.6

6. Compute probabilities from friend’s model

exp_orange <- exp(alpha_orange0 + alpha_orange1 * x) exp_apple <- exp(alpha_apple0 + alpha_apple1 * x)

prob_friend <- exp_orange / (exp_orange + exp_apple)

7. Friend’s predicted class

y_friend_pred <- ifelse(prob_friend > 0.5, “orange”, “apple”)

8. Compare both model predictions

agreement <- mean(y_your_pred == y_friend_pred) print(paste(“Fraction of agreement between models:”, round(agreement, 4)))

[1] “Fraction of agreement between models: 1”

plot(x, prob_your, col = ‘blue’, pch = 16, cex = 0.5, main = “Probability of Orange: Your Model vs Friend’s Model”, xlab = “x”, ylab = “Probability of Orange”) points(x, prob_friend, col = ‘red’, pch = 1, cex = 0.5) legend(“topright”, legend = c(“Your Model”, “Friend’s Model”), col = c(“blue”, “red”), pch = c(16, 1))