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 = 6350 and estimated standard deviation σ = 2750. 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 = 6350 and σ_x = 2750.

The probability distribution of x is approximately normal with μ_x = 6350 and σ_x = 1375.00.    

The probability distribution of x is not normal.

The probability distribution of x is approximately normal with μ_x = 6350 and σ_x = 1944.54.

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 decreased as n increased.

The probabilities stayed the same as n increased.    

The probabilities increased 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 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.

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 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 does not have leukopenia.

Answer 1

a)

µ =    6350          
σ =    2750          
              
P( X ≤    3500   ) = P( (X-µ)/σ ≤ (3500-6350) /2750)      
=P(Z ≤   -1.04   ) =   0.1500   (answer)

b)

std error = σ/√n=1944.54

answer: The probability distribution of x is approximately normal with μx = 6350 and σx = 1944.54.

Z =   (X - µ )/(σ/√n) = (   3500   -   6350.00   ) / (   2750.000   / √   2   ) =   -1.466  
                                          
P(X ≤   3500   ) = P(Z ≤   -1.466   ) =   0.0714                       (answer)

c)

std error = σ/√n=   1587.71

answer: The probability distribution of x is approximately normal with μx = 6350 and σx = 1587.71

Z =   (X - µ )/(σ/√n) = (   3500   -   6350.00   ) / (   2750.000   / √   3   ) =   -1.795  
                                          
P(X ≤   3500   ) = P(Z ≤   -1.795   ) =   0.0363                       (answer)

d)

The probabilities decreased as n increased.

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.