Rethinking Multiple-Choice

Sort these statements according to the topics.

Topic 1: Environmental Science
Topic 2: World History
Topic 3: Human Anatomy
Topic 4: Astrophysics
Topic 5: Literature Analysis

1. Statement: The Treaty of Versailles, signed in 1919, ended World War I and imposed heavy reparations and territorial losses on Germany.
2. Statement: The greenhouse effect is primarily caused by the accumulation of gases like carbon dioxide, methane, and nitrous oxide in the atmosphere.
3. Statement: Neurons are the fundamental units of the brain and nervous system, responsible for receiving sensory input from the external world, sending motor commands to our muscles, and transforming and relaying the electrical signals at every step in between.
4. Statement: Deforestation contributes to global warming as trees, which absorb carbon dioxide, are removed, increasing the amount of greenhouse gases in the atmosphere.
5. Statement: The Silk Road was an ancient network of trade routes that connected the East and West, playing a significant role in the cultural, economic, and political interactions between these regions.
6. Statement: Shakespeare's use of iambic pentameter in his plays adds a rhythmic quality to the dialogue, mirroring natural speech while allowing for varied emotional expression.
7. Statement: In George Orwell's "1984," the concept of Big Brother symbolizes the intrusive power of a totalitarian state, where the government exercises massive control over individuals' lives.
8. Statement: The human heart is a four-chambered organ that pumps blood throughout the body, with the left side pumping oxygenated blood and the right side pumping deoxygenated blood.
9. Statement: A black hole is a region in space where the gravitational pull is so strong that nothing, not even light, can escape from it.
10. Statement: The Big Bang Theory is a cosmological model that explains the early development of the Universe, suggesting it began from a very high-density and high-temperature state approximately 13.8 billion years ago.

# Model: Compare statement format to multiple-choice format
# Made by Noah Zimmermann (with ChatGPT)

library(tidyverse)

# Initialize parameters for tests
number_of_topics <- 5 # For statement test
number_of_options_per_question <- 4 # For multiple-choice test
set.seed(123) # For reproducibility

# Function to randomize learner attributes with a correlation to success
randomize_learner_attributes <- function(number_of_learners) {
  # Base attributes
  base_effort <- runif(number_of_learners, min = 1, max = 50)
  base_stress_performance <- runif(number_of_learners, min = 1, max = 50)
  base_correct_learning <- runif(number_of_learners, min = 1, max = 50)
  
  # Adjust attributes based on a simulated success probability
  success_probability <- runif(number_of_learners)
  effort <- base_effort + (success_probability * 50)
  stress_performance <- base_stress_performance + (success_probability * 50)
  correct_learning <- base_correct_learning + (success_probability * 50)
  
  tibble(
    LearnerID = 1:number_of_learners,
    Effort = effort,
    StressPerformance = stress_performance,
    CorrectLearning = correct_learning
  )
}

# Function to calculate proficiency
calculate_proficiency <- function(effort, stress_performance, correct_learning) {
  (effort + stress_performance + correct_learning) / 300
}

# Apply models and calculate mean probabilities
apply_models_and_calculate_means <- function(learners) {
  # Statement Test Model
  statement_results <- learners %>%
    mutate(Proficiency = calculate_proficiency(Effort, StressPerformance, CorrectLearning)) %>%
    mutate(Probability = 1 / number_of_topics + Proficiency * (1 - 1 / number_of_topics))
  
  # Multiple-Choice Test Model
  multiple_choice_results <- learners %>%
    mutate(Proficiency = calculate_proficiency(Effort, StressPerformance, CorrectLearning)) %>%
    mutate(
      SingleChoiceProbability = 1 / number_of_options_per_question + Proficiency * (1 - 1 / number_of_options_per_question),
      MultipleChoiceProbability = 1 - ((1 - 1 / number_of_options_per_question)^2) * (1 - Proficiency),
      
      
      MeanProbability = (SingleChoiceProbability + MultipleChoiceProbability) / 2
    )
  
# Determine success based on probability criteria
  
  statement_results <- statement_results %>%
    mutate(Successful = Probability >= 0.5)
  multiple_choice_results <- multiple_choice_results %>%
    mutate(Successful = MeanProbability >= 0.5)
  
# Calculate and print mean probabilities
  
  mean_statement_prob <- mean(statement_results$Probability)
  mean_multiple_choice_prob <- mean(multiple_choice_results$MeanProbability)
  
  list(StatementMeanProb = mean_statement_prob, MultipleChoiceMeanProb = mean_multiple_choice_prob,
       StatementSuccessRate = mean(statement_results$Successful),
       MultipleChoiceSuccessRate = mean(multiple_choice_results$Successful))
}

# Randomize learners based on a random number of learners and apply models

number_of_learners <- sample(15:60, 1)
learners <- randomize_learner_attributes(number_of_learners)
mean_probs_and_success <- apply_models_and_calculate_means(learners)

# Print mean probabilities and success rates

print(mean_probs_and_success)