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



Where a1 + a2 = a. If a1 = a2 = a/2, we have the usual 100(1 €“ a) % confidence interval for μ. In the above, When a1  a2 , the interval is not symmetric about μ. The length of the interval is L = σ(za1 + za2)/√n. Prove that the length of the interval L is minimized when a1 = a2 = a/2. Hint: Remember that Ф (za) = 1 €“ a, so Ф-1 (1 €“ a)= za, 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)].

X- Za, o/Vn sp Si + Za, o/Vn