Question

Given the following probabilities, find the value of a if Z ∼ N(0, 1)

(a) P(0 < Z < a) = 0.493

(b) P(−2.5 < Z < a) = 0.5

(c) P(Z < a) = 0.67

(d) P(Z > a) = 0.4025

Answer 1

for N(0, 1): $\mu=0,\sigma=1$

$z=\frac{x-\mu}{\sigma}$

a)

$P(0 < Z < a) =P( Z < a)-P(Z < 0)=P( Z < a)-0.5= 0.493$

$P( Z < a)=0.5+0.493=0.993$

$a=2.455$

b)

$P(-2.5 < Z < a) =P( Z < a)-P(Z <-2.5)=P( Z < a)-0.0062= 0.5$

$P( Z < a)=0.0062+0.5=0.5062$

$a=0.015$

c)

$P(Z < a) = 0.67$

$a=0.44$

d)

$P(Z > a) =1-P(Z < a) = 0.4025$

$P(Z < a)=1-0.4025=0.5975$

$a=0.245$