Question

Consider a two-sided confidence interval for the mean μ when σ is known;

X̄-Zα1 σ/√n ≤ μ ≤ X̄+Zα2 σ/√n

where α1 + α2 = α. if α1= α2 = α/2, we have the usual 100(1-α)% confidence interval for μ. In the above, when α1 ≠ α2 , the interval is not symmetric about μ. Prove that the length of the interval L is minimized when α1= α2= α/2. Hint remember that Φ Zα = (1- α) , so Φ^-1 (1- α) = Zα, and the relationship between the derivative of a function y =f(x) and the inverse x=f^-1 (y) is (d/dy)f^-1 (y) = 1/ [d/dx f(x)] .

Answer 1