Question

Given that x is a normal variable with mean μ = 48 and standard deviation σ = 6.3, find the following probabilities. (Round your answers to four decimal places.) (

a) P(x ≤ 60)

(b) P(x ≥ 50)

(c) P(50 ≤ x ≤ 60)

Please break down explanation a-c.

Answer 1

Given

X is a Normal variable

mean (μ) = 48

standard deviation (σ) = 6.3

First convert into Z score which is given as

When x= 60

Z =1.90

Now,

P(x ≤ 60) = P(Z ≤ 1.90)

From Z-score table,

P(Z ≤ 1.90) = 0.9713

b)

When x= 50

Z = 0.32

Now,

P(x ≥ 50) = P(Z ≥ 0.32) = 1 - P(Z ≤ 0.32)

From Z-score table, ( table only give the area under the curve below the Z score )

P(Z ≤ 0.32) = 0.6255

Therefore,

P(x ≥ 50) = P(Z ≥ 0.32) = 1 - P(Z ≤ 0.32) =1 - 0.6255 = 0.3745

c)

When x= 50

Z = 0.32

When x= 60

Z =1.90

Now,

P(50 ≤ x ≤ 60) = P(x ≤ 60) - P(x ≤ 50)

=P(Z ≤ 1.90) - P(Z ≤ 0.32)

= 0.9713 - 0.6255

= 0.3458