Question

Compute each probability or quantile.

(a) X ∼ N(3, 0.0225), P(X ≤ 3.25).

(b) X ∼ N(52, 49), P(X ≥ 60).

(c) X ∼ N(3.7, 4.55), P(3.0 ≤ X ≤ 10.0).

(d) X ∼ N(25, 36). Find the first and third quartiles for X.

Answer 1

Part a

We have to find P(X ≤ 3.25)

Z = (X - µ)/σ

µ = 3

σ^2 = 0.0255

σ = sqrt(0.0225) = 0.15

Z = (3.25 - 3)/0.15

Z = 1.666667

P(Z<1.666667) = P(X ≤ 3.25) = 0.95221

(by using z-table)

Required probability = 0.95221

Part b

We have to find P(X ≥ 60) = 1 – P(X<60)

Z = (X - µ)/σ

µ = 52

σ^2 = 49

σ = sqrt(49) = 7

Z = (60 - 52)/7

Z = 1.142857

P(Z<1.142857) = P(X<60) = 0.873451

(by using z-table)

Required probability = 0.873451

Part c

P(3.0 ≤ X ≤ 10.0) = P(X<10) – P(X<3)

Z = (X - µ)/σ

µ = 3.7

σ^2 = 4.55

σ = 2.133073

Find P(X<10)

Z = (10 - 3.7)/ 2.133073

Z = 2.953485

P(Z<2.953485) = P(X<10) = 0.998429

(by using z-table)

Now find P(X<3)

Z = (3 - 3.7)/ 2.133073

Z = -0.32817

P(Z<-0.32817) = P(X<3) = 0.371393

(by using z-table)

P(3.0 ≤ X ≤ 10.0) = P(X<10) – P(X<3)

P(3.0 ≤ X ≤ 10.0) = 0.998429 - 0.371393

P(3.0 ≤ X ≤ 10.0) = 0.62704

Required probability = 0.62704

Part d

X = µ + Z*σ

µ = 25

σ^2 = 36

σ = 6

Z for first quartile = -0.67449

Z for third quartile = 0.67449

First quartile = 25 - 0.67449*6 = 20.9531

First quartile = 20.9531

Third quartile = 25 + 0.67449*6 =29.0469

Third quartile = 29.0469