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 = 6150and 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 not normal.The probability distribution of x is approximately normal with μ_x = 6150 and σ_x = 1555.63.     The probability distribution of x is approximately normal with μ_x = 6150 and σ_x = 2200.The probability distribution of x is approximately normal with μ_x = 6150 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 decreased as n increased.The probabilities increased as n increased.     The probabilities stayed the same 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 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.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

Given

It is given that white blood cell counts are approximately normally distributed

(a) P(x < 3500)

Since, it is given that the

For the given values, Z is calculated as below (since it is normally distributed)

Hence, the given sample size is 1, n = 1. Therefore, the z value for x = 3500 is given as

Hence, P(X < 3500) = P(Z < -1.2045). The probability value can be estinmated from the standard normal distribution using Excel with the function NORMSDIST. The final function is given as NORMSDIST(-1.2045)

The answer is given as 0.1142

Therefore, the probability that the x < 3500 is given as 0.1142

(b) For the same mean and standard deviation with the sample size of 2 that is n = 2 the distribution is given as normally distributed with mean = 6150. Becuase for normally distributed data the popuilation mean will be approximately same as sample mean. And the standard error is given as /. Hence, the standard devation is given as 1555.63. Hence, the answer final answer is The probability distribution of x is approximately normal with μx = 6150 and σx = 1555.63.     

Updated Probability of x < 3500 is given as

Hence, P(X < 3500) = P(Z < -1.7035). The probability value can be estinmated from the standard normal distribution using Excel with the function NORMSDIST. The final function is given as =NORMSDIST(-1.7035)

The answer is given as

Therefore, the probability that the x < 3500 is given as 0.0442

(c) Probability of x < 3500 for n = 3

Hence, P(X < 3500) = P(Z < -2.0863). The probability value can be estinmated from the standard normal distribution using Excel with the function NORMSDIST. The final function is given as NORMSDIST(-2.0863)

The answer is given as 0.0185

Therefore, the probability that the x < 3500 is given as 0.0185

(d)

The probabilities for n = 1, 2 and 3 are given as 0.1142, 0.0442 and 0.0185 respectively. It can be noticed that as the n increases the probability decreases. Hence, the ansewr is given as The probabilities decreased as n increased.

The final conclusion is given as

Since the probability decreased for increased number of sample size, it can be concluded that The person probably does not have leukopenia