Question

Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is approximately normal, with mean μ = 6650 and estimated standard deviation σ = 2200. A test result of x < 3500 is an indication of leukopenia. This indicates bone marrow depression that may be the result of a viral infection.

(a) What is the probability that, on a single test, x is less than 3500? (Round your answer to four decimal places.)

(b) Suppose a doctor uses the average x for two tests taken about a week apart. What can we say about the probability distribution of x?

The probability distribution of x is approximately normal with μ_x = 6650 and σ_x = 1555.63.

The probability distribution of x is approximately normal with μ_x = 6650 and σ_x = 2200.    

The probability distribution of x is not normal.

The probability distribution of x is approximately normal with μ_x = 6650 and σ_x = 1100.00.

What is the probability of x < 3500? (Round your answer to four decimal places.)


(c) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to four decimal places.)


(d) Compare your answers to parts (a), (b), and (c). How did the probabilities change as n increased?

The probabilities stayed the same as n increased.

The probabilities increased as n increased.    

The probabilities decreased as n increased.

If a person had x < 3500 based on three tests, what conclusion would you draw as a doctor or a nurse?

It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia.

It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.    

It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.

It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia.

Answer 1

a)

µ =    6650          
σ =    2200          
              
P( X < 3500   ) = P( (X-µ)/σ ≤ (3500-6650) /2200)      
=P(Z < -1.43   ) =   0.0761   (answer)

b)

µ=   6650
σ=   2200
n =    2

std error = σ/√n=   1555.63

The probability distribution of x is approximately normal with μ_x = 6650 and σ_x = 1555.63.

Z =   (X - µ )/(σ/√n) = (   3500   -   6650.00   ) / (   2200.000   / √   2   ) =   -2.025  
                                          
P(X ≤   3500   ) = P(Z ≤   -2.025   ) =   0.0214                       (answer)

c)

µ =    6650                                      
σ =    2200                                      
n=   3                                      
                                          
X =   3500                                      
                                          
Z =   (X - µ )/(σ/√n) = (   3500   -   6650.00   ) / (   2200.000   / √   3   ) =   -2.480  
                                          
P(X ≤   3500   ) = P(Z ≤   -2.480   ) =   0.0066                       (answer)

d)

The probabilities decreased as n increased.

e)

It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.