Let 0   . Then a 100(1  )% CI for when n is large is

x

 z


s n

, x z


s n



The choice
/2 yields the usual interval derived in Section 7.2; if /2, this interval is not symmetric about x

. The width of this interval is w
s(z z )/n.

Show that w is minimized for the choice
/2, so that the symmetric interval is the shortest. [Hints:

(a) By definition of z, (z)
1  , so that z
1

(1  );