Question

Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. (Round your answers to four decimal places.)

(a)    P(0 ≤ Z ≤ 2.72)

(b)    P(0 ≤ Z ≤ 1)

(c)    P(−2.90 ≤ Z ≤ 0)

(d)    P(−2.90 ≤ Z ≤ 2.90)

(e)    P(Z ≤ 1.37)
(f)    P(−1.55 ≤ Z)
(g)    P(−1.90 ≤ Z ≤ 2.00)
(h)    P(1.37 ≤ Z ≤ 2.50)
(i)    P(1.90 ≤ Z)

(j)    P(|Z| ≤ 2.50)

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

Part a)

P ( 0 <= Z <= 2.72 ) = P ( Z < 2.72 ) - P ( Z < 0 )
P ( 0 <= Z <= 2.72 ) = 0.9967 - 0.5
P ( 0 <= Z <= 2.72 ) = 0.4967

Part b)

P ( 0 <= Z <= 1 ) = P ( Z < 1 ) - P ( Z < 0 )
P ( 0 <= Z <= 1 ) = 0.8413 - 0.5
P ( 0 <= Z <= 1 ) = 0.3413

Part c)

P ( -2.9 <= Z <= 0 ) = P ( Z < 0 ) - P ( Z < -2.9 )
P ( -2.9 <= Z <= 0 ) = 0.5 - 0.0019
P ( -2.9 <= Z <= 0 ) = 0.4981

Part d)

P ( -2.9 <= Z <= 2.9 ) = P ( Z < 2.9 ) - P ( Z < -2.9 )
P ( -2.9 <= Z <= 2.9 ) = 0.9981 - 0.0019
P ( -2.9 <= Z <= 2.9 ) = 0.9963

Part e)

P ( Z <= 1.37 ) = 0.9147

Part f)

P ( Z >= -1.55 ) = 1 - P ( Z < -1.55 )
P ( Z >= -1.55 ) = 1 - 0.0606
P ( Z >= -1.55 ) = 0.9394

Part g)

P ( -1.9 <= Z <= 2 ) = P ( Z < 2 ) - P ( Z < -1.9 )
P ( -1.9 <= Z <= 2 ) = 0.9772 - 0.0287
P ( -1.9 <= Z <= 2 ) = 0.9485

Part h)

P ( 1.37 <= Z <= 2.5 ) = P ( Z < 2.5 ) - P ( Z < 1.37 )
P ( 1.37 <= Z <= 2.5 ) = 0.9938 - 0.9147
P ( 1.37 <= Z <= 2.5 ) = 0.0791

Part i)

P ( Z >= 1.9 ) = 1 - P ( Z < 1.9 )
P ( Z >= 1.9 ) = 1 - 0.9713
P ( Z >= 1.9 ) = 0.0287

Part j)

P ( -2.5 < Z < 2.5 ) = P ( Z < 2.5 ) - P ( Z < -2.5 )
P ( -2.5 < Z < 2.5 ) = 0.9938 - 0.0062
P ( -2.5 < Z < 2.5 ) = 0.9876