(i) If () denotes the standard normal density and Z N(0, 1), then for any t...

(i) If φ(·) denotes the standard normal density and Z ∼ N(0, 1), then for any t > 0,

(

1 t − 1 t 3 )φ(t) < P{Z ≥ t} ≤ φ(t)

t . (

Prove the right-hand inequality.
(ii) Prove the left inequality in (13.64). Hint: Feller (1968) p.179 notes the negative of the derivative of the left side ( 1 t − 1 t 3 )φ(t) is equal to (1 − 3t−4)φ(t), which is certainly less than φ(t).
(iii) Use (13.64) to show that, for any fixed α, any δ > 0, and all large enough n:

(1 − δ)2 log n ≤ z1− α
n ≤ 
2 log n . (13.65)