Let 0 # g # a. Then a 100(1 2 a)% CI for m when n islarge is1x 2 zg ? sn, x 1 za2g ?
Chapter 7, Problem 60(choose chapter or problem)
Let 0 # g # a. Then a 100(1 2 a)% CI for m when n islarge is1x 2 zg ? sn, x 1 za2g ? sn2The choice g 5 ay2 yields the usual interval derived inSection 7.2; if g ay2, this interval is not symmetric aboutx#. The width of this interval is w 5 s(zg 1 za2g)yn.Show that w is minimized for the choice g 5 ay2, sothat the symmetric interval is the shortest. [Hints: (a) Bydefinition of za, F(za) 5 1 2 a, so that za 5 F21(1 2 a);(b) the relationship between the derivative of a functiony 5 f(x) and the inverse function x 5 f21(y) is(dydy) f 21(y) 5 1yf9(x).]
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