Question

Suppose x is a normally distributed random variable with µ = 10.1 and σ = 3.4. Find each of the following probabilities: a. P(5.7 ≤ X ≤ 16.2)= ________ b. P(5.6 ≤ X ≤ 15.6)= ________ c. P(11.5 ≤ X ≤ 14.1)= ________ d. P(X ≥ 10.7)=________ e. P(X ≤ 14.4)=________

Answer 1

Answer:

Step-by-step explanation:

The formula for normal distribution is

z = (x - µ)/σ

a) P(5.7 ≤ X ≤ 16.2)

For x = 5.7,

z = (5.7 - 10.1)/3.4 = - 1.3

The corresponding probability from the normal distribution table is 0.0968

For x = 16.2,

z = (16.2 - 10.1)/3.4 = 1.8

The corresponding probability from the normal distribution table is

0.9641

Therefore,

P(5.7 ≤ X ≤ 16.2) = 0.9641 - 0.0968 =

0.8673

b) P(5.6 ≤ X ≤ 15.6)

For x = 5.6,

z = (5.6 - 10.1)/3.4 = - 1.32

The corresponding probability from the normal distribution table is 0.09342

For x = 15.6

z = (15.6 - 10.1)/3.4 = 1.62

The corresponding probability from the normal distribution table is

0.9474

Therefore,

P(5.6 ≤ X ≤ 15.6) = 0.9474 - 0.09342 = 0.85398

c) P(11.5 ≤ X ≤ 14.1)

For x = 11.5,

z = (11.5 - 10.1)/3.4 = 0.41

The corresponding probability from the normal distribution table is 0.6591

For x = 14.1

z = (14.1 - 10.1)/3.4 = 1.18

The corresponding probability from the normal distribution table is

0.881

Therefore,

P(11.5 ≤ X ≤ 14.1) = 0.881 - 0.6591 = 0.2219

d) P(X ≥ 10.7) = 1 - P(X ≤ 10.7)

For x = 10.7

z = (10.7 - 10.1)/3.4 = 0.18

The corresponding probability from the normal distribution table is

0.5714

Therefore,

P(X ≥ 10.7) = 1 - 0.5714 = 0.4286

e) P(X ≤ 14.4)

For x = 14.4

z = (14.4 - 10.1)/3.4 = 1.26

The corresponding probability from the normal distribution table is

0.9131

Therefore

P(X ≤ 14.4) = 0.9131