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 = 42 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

Since the distribution is normal with mean, mean μ = 42 and Standard deviation, σ = 14 so, Z statistic is applicable here for probability calculation hence,

a) P(X<60) is calculated by finding the Z score at 60

The desired probability is calculated by the excel tool or by table given below

So, P(X<60)=P(Z<1.286)

=1-0.0992,

=0.9008

b) P(X>16) is calculated similarly as calculated in part a.

=P(X>16)=P(Z>1.857), probability calculated by the graph shown below,

=0.9683

c) x is between 16 and 60

P(16<X<60)

Z at X=16

and at X=60

P(16<X<60)=P(-1.857<Z<1.286) now by graph shown below

=0.9008-0.0317

=0.8691

d) x is more than 60

P(X>60) so by Z

Hence P(X>60)=P(Z>1.286)

=0.0992