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# Define the z-values for the given probabilities
z_alpha = qnorm(0.10, lower.tail = FALSE) # z-value for alpha = 0.10
z_beta = qnorm(0.95)  # z-value for beta = 0.95

# Define the probabilities from the problem
p1 = 1/2
p2 = 2/3

# Solve for n using the equation derived from setting the two expressions for c equal to each other
n = ((z_alpha * sqrt(p1*(1-p1)) + p1) - (z_beta * sqrt(p2*(1-p2)) + p2))^2 / (p2 - p1)^2
n = (8.50)^2

# Solve for c using one of the equations for c
c = z_alpha * sqrt(n*p1*(1-p1)) + n*p1

# Output the calculated values before rounding
cat("Calculated sample size (n) before rounding:", n, "\n")
cat("Calculated critical value (c) before rounding:", c, "\n")

# Round n and c to be conservative as per the given solution
n_rounded = ceiling(n) # Conservative rounding for n
c_rounded = floor(c) # Conservative rounding for c

# Output the rounded values
cat("Sample size (n) after conservative rounding:", n_rounded, "\n")
cat("Critical value (c) after conservative rounding:", c_rounded, "\n")
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