Chapter 6.5, Problem 16P

Medical: White Blood Cells 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 = 7500 and estimated standard deviation σ = 1750 (see reference in Problem 15). 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.

1. (a) What is the probability that, on a single test, x is less than 3500?
2. (b) Suppose a doctor uses the average $\bar{x}$ for two tests taken about a week apart. What can we say about the probability distribution of $\bar{x}$? What is the probability of $\bar{x}<\text{35}00$?
3. (c) Repeat part (b) for n = 3 tests taken a week apart.
4. (d) Interpretation Compare your answers to parts (a), (b), and (c). How did the probabilities change as n increased? If a person had $\bar{x}<\text{35}00$ based on three tests, what conclusion would you draw as a doctor or a nurse?

15. Medical: Blood Glucose Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12-hour fast. Assume that for people under 50 years old, x has a distribution that is approximately normal, with mean μ = 85 and estimated standard deviation σ = 25 (based on information from Diagnostic Tests with Nursing Applications, edited by S. Loeb, Springhouse). A test result x < 40 is an indication of severe excess insulin, and medication is usually prescribed.

1. (a) What is the probability that, on a single test, x < 40?
2. (b) Suppose a doctor uses the average $\bar{x}$ for two tests taken about a week apart. What can we say about the probability distribution of $\bar{x}$? Hint: See Theorem 6.1. What is the probability that $\bar{x}<40$?
3. (c) Repeat part (b) for n = 3 tests taken a week apart.
4. (d) Repeat part (b) for n = 5 tests taken a week apart.
5. (e) Interpretation Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased? Explain what this might imply if you were a doctor or a nurse. If a patient had a test result of $\bar{x}<40$ based on five tests, explain why either you are looking at an extremely rare event or (more likely) the person has a case of excess insulin.

THEOREM 6.1 For a Normal Probability Distribution Let x be a random variable with a normal distribution whose mean is μ and whose standard deviation is σ. Let $\bar{x}$ be the sample mean corresponding to random samples of size n taken from the x distribution. Then the following are true:

1. (a) The $\bar{x}$ distribution is a normal distribution.
2. (b) The mean of the $\bar{x}$ distribution is μ.
3. (c) The standard deviation of the $\bar{x}$ distribution is $\sigma /\sqrt{n}$.

We conclude from Theorem 6.1 that when x has a normal distribution, the $\bar{x}$ distribution will be normal for any sample size n. Furthermore, we can convert the $\bar{x}$ distribution to the standard normal z distribution using the following formulas.

$\begin{array}{l}{\mu }_{\bar{x}}=\mu \\ {\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}\\ z=\frac{\bar{x}-{\mu }_{\bar{x}}}{{\sigma }_{\bar{x}}}=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}\end{array}$

where n is the sample size,

μ is the mean of the x distribution, and

σ is the standard deviation of the x distribution.

Theorem 6.1 is a wonderful theorem! It states that the $\bar{x}$ distribution will be normal provided the x distribution is normal. The sample size n could be 2, 3, 4, or any other (fixed) sample size we wish. Furthermore, the mean of the $\bar{x}$ distribution is μ (same as for the x distribution), but the standard deviation is $\sigma \text{/}\sqrt{n}$ (which is, of course, smaller than σ). The next example illustrates Theorem 6.1.