Find the value of a so that if Z is a standard normal random

Find the value of a so that, if Z is a standard normal random variable, P(a < = Z <= 0.54) = 0.3898.

Solution

P(a < = Z <= 0.54) = 0.3898 =>      P(Z <= 0.54) - P(Z<=a) = 0.3898

By normal table we have P(Z<=0.54) = 0.7054, thus:

0.7054 - P(Z<=a) = 0.3898   =>   P(Z<=a) = 0.7054 - 0.3898 = 0.3156

Again by inverse normal table: a = -0.48 when P(Z<=a) = 0.3156

Therefore a = -0.48