In: Statistics and Probability
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 = 6100 and estimated standard deviation σ = 2000. 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? Pick one
-The probability distribution of x is approximately normal with μx = 6100 and σx = 1000.00.
-The probability distribution of x is approximately normal with μx = 6100 and σx = 2000.
-The probability distribution of x is not normal.
-The probability distribution of x is approximately normal with μx = 6100 and σx = 1414.21.
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? Pick one
-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? Pick one
-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.
-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.
Solution :
Given that ,
mean = = 6100
standard deviation = = 2000
a) P(x < 3500) = P[(x - ) / < (3500 - 6100) / 2000]
= P(z < -1.30)
Using z table,
= 0.0968
b) n = 2
= = 6100
= / n = 2000/ 2 = 1414.21
The probability distribution of x is approximately normal with μx = 6100 and σx = 1414.21
P( < 3500) = P(( - ) / < (3500 - 6100) / 1414.21)
= P(z < -1.84)
Using z table
= 0.0329
c) n = 3
= = 6100
= / n = 2000/ 3 = 1154.70
The probability distribution of x is approximately normal with μx = 6100 and σx = 1154.70
P( < 3500) = P(( - ) / < (3500 - 6100) / 1154.70)
= P(z < -2.25)
Using z table
= 0.0122
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