CI_simulation

# Set the parameters
set.seed(123)  # for reproducibility
sample_size <- 100  # sample size
num_simulations <- 1000  # number of simulations
true_mean <- 50  # true population mean
standard_deviation <- 4  # standard deviation

# Function to perform one simulation
perform_simulation <- function() {
  # Generate a random sample from N(mu, sigma^2)
  sample_data <- rnorm(sample_size, true_mean, standard_deviation)
  
  # Calculate the sample mean and standard error
  sample_mean <- mean(sample_data)
  standard_error <- sd(sample_data) / sqrt(sample_size)
  
  # Calculate the 90% confidence interval
  lower_bound <- sample_mean - qnorm(0.95) * standard_error
  upper_bound <- sample_mean + qnorm(0.95) * standard_error
  
  # Check if the true mean is within the confidence interval
  contains_true_mean <- lower_bound <= true_mean && upper_bound >= true_mean
  
  return(contains_true_mean)
}

# Perform multiple simulations
contains_true_mean_count <- sum(replicate(num_simulations, perform_simulation()))

# Calculate the proportion of simulations where the true mean is within the confidence interval
proportion_contains_true_mean <- contains_true_mean_count / num_simulations

# Output the result
cat("Proportion of simulations where the true mean is within the confidence interval:", proportion_contains_true_mean)

## Proportion of simulations where the true mean is within the confidence interval: 0.914

library(ggplot2)

# Set the parameters
set.seed(123)  # for reproducibility
sample_size <- 100  # sample size
num_simulations <- 100  # number of simulations
true_mean <- 50  # true population mean
standard_deviation <- 4  # standard deviation

# Function to perform one simulation
perform_simulation <- function() {
  # Generate a random sample from N(mu, sigma^2)
  sample_data <- rnorm(sample_size, true_mean, standard_deviation)
  
  # Calculate the sample mean and standard error
  sample_mean <- mean(sample_data)
  standard_error <- sd(sample_data) / sqrt(sample_size)
  
  # Calculate the confidence interval
  lower_bound <- sample_mean - qnorm(0.95) * standard_error
  upper_bound <- sample_mean + qnorm(0.95) * standard_error
  
  return(data.frame(lower_bound = lower_bound, upper_bound = upper_bound, true_mean = true_mean))
}

# Perform multiple simulations
simulation_results <- lapply(1:num_simulations, function(x) perform_simulation())

# Convert the results to a data frame
simulation_df <- do.call(rbind, simulation_results)

# Plot the results
ggplot(simulation_df, aes(x = true_mean, y = as.factor(seq_along(true_mean)))) +
  geom_segment(aes(x = lower_bound, xend = upper_bound, y = as.factor(seq_along(true_mean)), yend = as.factor(seq_along(true_mean))), color = "blue", size = 1) +
  geom_vline(xintercept = true_mean, linetype = "dashed", color = "red", size = 1) +
  labs(title = "Simulated 90% Confidence Intervals",
       x = "True Mean",
       y = "") +
  theme_minimal() +
  scale_y_discrete(labels = NULL) +
  coord_cartesian(xlim = c(min(simulation_df$lower_bound), max(simulation_df$upper_bound)))

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.