Question

Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy circles, x is approximately normally distributed with mean μ = 44 and standard deviation σ = 14. Find the following probabilities. (Round your answers to four decimal places.)

(a) x is less than 60


(b) x is greater than 16


(c) x is between 16 and 60


(d) x is more than 60 (This may indicate an infection, anemia, or another type of illness.)

Answer 1

a)

Here, μ = 44, σ = 14 and x = 60. We need to compute P(X <= 60). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (60 - 44)/14 = 1.14

Therefore,
P(X <= 60) = P(z <= (60 - 44)/14)
= P(z <= 1.14)
= 0.8729

b)
Here, μ = 44, σ = 14 and x = 16. We need to compute P(X >= 16). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (16 - 44)/14 = -2

Therefore,
P(X >= 16) = P(z <= (16 - 44)/14)
= P(z >= -2)
= 1 - 0.0228 = 0.9772

c)

Here, μ = 44, σ = 14, x1 = 16 and x2 = 60. We need to compute P(16<= X <= 60). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (16 - 44)/14 = -2
z2 = (60 - 44)/14 = 1.14

Therefore, we get
P(16 <= X <= 60) = P((60 - 44)/14) <= z <= (60 - 44)/14)
= P(-2 <= z <= 1.14) = P(z <= 1.14) - P(z <= -2)
= 0.8729 - 0.0228
= 0.8501

d)

Here, μ = 44, σ = 14 and x = 60. We need to compute P(X >= 60). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (60 - 44)/14 = 1.14

Therefore,
P(X >= 60) = P(z <= (60 - 44)/14)
= P(z >= 1.14)
= 1 - 0.8729 = 0.1271