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 = 35 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

Answer 1

Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood and follows

N( 35,196 )

We'll first convert X into standard normal variate then find the value of Z from normal probability table or from excel easily.

a) P( X < 60 ) = P ( ( X - 35)/14 < ( 60 - 35 )/14)

= P( Z < 25/14) = P ( Z < 1.785 )

= 0.9629

b) P ( X > 16) = P ( Z > (16-35)/14) = P( Z > - 1.357 )

= 0.9126

c ) P ( 16 <X < 60 ) = P ( (16-35)/14 < Z < (60 - 35)/14 )

= P( -1.357 < Z < 1.785 )

= P ( Z < 1.785 ) - P( Z < -1.357)

= 0.9629 - (1 - P(Z > - 1.357))

= 0.9629 - 0.0874 = 0.8755

d) P ( X > 60 ) = P ( Z > 1.785)

= 1- P( Z < 1.785)

= 1 - 0.9629 = 0.0371