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# Calculate the z-values for the corresponding confidence levels
z_alpha = qnorm(0.10, lower.tail = FALSE)  # For the 90th percentile
z_beta = qnorm(0.05, lower.tail = TRUE)    # For the 95th percentile (one-tailed test)

# Solve for n using the algebraic equation derived from the normal approximation
n = ((z_alpha * sqrt(0.25) + z_beta * sqrt(2/9)) / (2/3 - 1/2))^2

# Conservative rounding of n
n_conservative = ceiling(n)

# Solve for c using the value of n
c = z_alpha * sqrt(n_conservative * 0.25) + 0.5 * n_conservative

# Conservative rounding of c
c_conservative = floor(c)

# Print the results
cat("Calculated sample size n:", n, "\n")
cat("Conservative rounded n:", n_conservative, "\n")
cat("Calculated critical value c:", c, "\n")
cat("Conservative rounded c:", c_conservative, "\n")