Week9Discussion

Exercise #4 page 354

Write a program to find the average of 1000 random digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Have the program test to see if the average lies within three standard deviations of the expected value of 4.5. Modify the program so that it repeats this simulation 1000 times and keeps track of the number of times the test is passed. Does your outcome agree with the Central Limit Theorem?

# Function to perform simulation and test
average_test <- function() {
  digits <- sample(0:9, 1000, replace = TRUE)
  average <- mean(digits)
  expected_value <- 4.5
  std_dev <- sd(digits)

  lower_bound <- expected_value - 3 * std_dev
  upper_bound <- expected_value + 3 * std_dev

  test_result <- average >= lower_bound && average <= upper_bound
  
  return (test_result)
}

# Function to repeat simulation and test 1000 times
run_simulation <- function() {
  passed_tests <- 0
  for (i in 1:1000) {
    if (average_test()) {
      passed_tests <- passed_tests + 1
    }
  }
  return(passed_tests)
}

# Main function
main <- function() {
  passed_tests <- run_simulation()
  cat("Number of times the test is passed:", passed_tests, "\n")
}

# Run the main function
main()

## Number of times the test is passed: 1000

Since all tests passed, meaning all averages lie within three standards deviation from the expected value (\(4.5\)), then the outcome agree with the Central Limit Theorem.