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 = 39 and standard deviation σ = 13. 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

Solution :

Given that ,

mean = = 39

standard deviation = = 13

(a)

P(x < 60) = P((x - ) / < (60 - 39) / 13)

= P(z < 1.6154)

Using standard normal table,

P(x < 16) = 0.9469

Probability = 0.9469

(b)

P(x > 16) = 1 - P(x < 16 )

= 1 - P((x - ) / < (16 - 39) / 13)

= 1 - P(z <-1.7692)

= 1 - 0.0384   

= 0.9616

P(x > 16) = 0.9616

Probability = 0.9616

(c)

P(16 < x < 60) = P((16 - 39)/ 13) < (x - ) / < (60 - 39) / 13) )

= P(-1.7692 < z < 1.6154)

= P(z < 1.6154) - P(z < -1.7692)

= 0.9469 - 0.0384

Probability = 0.9085

(d)

P(x > 60) = 1 - P(x < 60)

= 1 - P((x - ) / < (60 - 39) / 13)

= 1 - P(z < 1.6154)

= 1 - 0.9469   

= 0.0531

P(x > 60) = 0.0531

Probability = 0.0531