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 = 6900 and estimated standard deviation σ = 2100. 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 = 6900 and σ_x = 1050.00.

The probability distribution of x is not normal.  

  The probability distribution of x is approximately normal with μ_x = 6900 and σ_x = 2100.

The probability distribution of x is approximately normal with μ_x = 6900 and σ_x = 1484.92.

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 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.

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.

Answer 1

a Here x has a distribution that is approximately normal, with mean μ = 6900 and estimated standard deviation σ = 2100

Now we need to find

As distribution is normal we can convert x to z

b. As population is normal, sample mean is also normal with mean=6900 and standard deviation=2100/sqrt(2)=1484.92

So answer is

The probability distribution of x is approximately normal with μ_x = 6900 and σ_x = 1484.92.

Now

c. Now for n=3 standard deviation is 2100/sqrt(3)=1212.44

Now we need to find

d.

The probabilities decreased as n increased.

As probability for two or three is less than 0.05, we conclude test is significant

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.