Question

Medical: Blood Protoplasm Porphyrin is a pigment in the 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 adults, x is approximately normally distributed with mean = 38 and standard deviation = 12. What is the probability that 20 healthy adults will have the average porphyrin level ... a) less than 34? b) greater than 42? c) between 34 and 42?

Answer 1

a)

P ( X̅ < 34) = ?

X ~ N ( µ = 38 , σ = 12 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 34 - 38 ) / ( 12 / √20 )
Z = -1.49
P ( ( X - µ ) / ( σ/√(n)) = ( 34 - 38 ) / ( 12 / √(20) )
P ( X̅ < 34) = P ( Z < -1.49 )
= 0.0681

b)

P ( X > 42 ) = 1 - P ( X < 42 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 42 - 38 ) / ( 12 / √ ( 20 ) )
Z = 1.49
P ( ( X - µ ) / ( σ / √ (n)) > ( 42 - 38 ) / ( 12 / √(20) )
P ( Z > 1.49 )
P ( X̅ > 42 ) = 1 - P ( Z < 1.49 )
P ( X̅ > 42 ) = 1 - 0.9319
P ( X̅ > 42 ) = 0.0681

c)

P ( 34 < X̅ < 42 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 34 - 38 ) / ( 12 / √(20))
Z = -1.49
Z = ( 42 - 38 ) / ( 12 / √(20))
Z = 1.49
P ( 34 < X̅ < 42 ) = P ( Z < 1.49 ) - P ( Z < -1.49 )
P ( 34 < X̅ < 42 ) = 0.9319 - 0.0681
P ( 34 < X̅ < 42 ) = 0.8638