Question

Find the following probabilities:

a) Pr{Z < 0.66}

b) Pr{Z ≥ -0.66}

c) Pr{-2.01 < Z < 2.01}

d) Pr{-1.91 < Z < 0.0}

e) Pr{Z < -1.35 or Z > 1.35} (you want the probability that Z is outside the range -1.03 to 1.03)

Answer 1

a) $P(Z<0.66) = \Phi(0.66) = 0.745373$

b) $P(Z\geq -0.66) = 1 - P(Z<-0.66)=1- \Phi(-0.66) = 1 - 0.254627 = 0.745373$

c)

$P(-2.01<Z<2.01) = P(Z<2.01) - P(Z<-2.01)$

$\Rightarrow P(-2.01<Z<2.01) = \Phi(2.01) - \Phi(-2.01)$

$\Rightarrow P(-2.01<Z<2.01) = [1 - \Phi(-2.01)] - \Phi(-2.01)$

$\Rightarrow P(-2.01<Z<2.01) = (1 - 0.0222156) - 0.0222156$

$\Rightarrow P(-2.01<Z<2.01) = 0.955569$

d)

$P(-1.91 <Z<0) = P(Z<0) - P(Z<-1.91) =\Phi(0)-\Phi(-1.91)$

$\Rightarrow P(-1.91 <Z<0) = 0.5 - 0.0280666$

$\Rightarrow P(-1.91 <Z<0) = 0.471933$

e)

$P(Z<-1.35,Z>1.35) = P(Z<-1.35) &plus; P(Z>1.35)$

$\Rightarrow P(Z<-1.35,Z>1.35) = 2*P(Z<-1.35)$

$\Rightarrow P(Z<-1.35,Z>1.35) = 2*0.0885080$

$\Rightarrow P(Z<-1.35,Z>1.35) = 0.177016$