Statistical Inference Course Question 5(a)

# Set the p-value and sample size n
p_value <- 0.05
n_values <- 1:20

# Initialize vectors to store x and y values 
x_values <- numeric(length(n_values))
likelihood_ratios <- numeric(length(n_values))

# Loop over each n 
for (n in n_values) {
  # Calculate the critical value x 
  x <- qnorm(1 - p_value, mean = 0, sd = sqrt(1/n))
  
  # Calculate the likelihood ratio
  lr <- dnorm(x, mean = 0, sd = sqrt(1/n)) / dnorm(x, mean = 1, sd = sqrt(1/n))
  likelihood_ratios[n] <- lr
}

# Plot the likelihood ratios on a log scale vs sample size
plot(n_values, likelihood_ratios, type = "b", log = "y", xlab = "Sample size n", ylab = "Likelihood Ratio (log scale)", main = "Likelihood Ratio vs. Sample Size")