Question

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 = 52 and estimated standard deviation σ = 25. A test result x < 40 is an indication of severe excess insulin, and medication is usually prescribed.

(a) What is the probability that, on a single test, x < 40? (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? Hint: See Theorem 6.1.

The probability distribution of x is approximately normal with μ_x = 52 and σ_x = 25.The probability distribution of x is approximately normal with μ_x = 52 and σ_x = 12.50.     The probability distribution of x is approximately normal with μ_x = 52 and σ_x = 17.68.The probability distribution of x is not normal.

What is the probability that x < 40? (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) Repeat part (b) for n = 5 tests taken a week apart. (Round your answer to four decimal places.)


(e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased?

YesNo     

Explain what this might imply if you were a doctor or a nurse.

The more tests a patient completes, the weaker is the evidence for excess insulin.The more tests a patient completes, the stronger is the evidence for excess insulin.     The more tests a patient completes, the stronger is the evidence for lack of insulin.The more tests a patient completes, the weaker is the evidence for lack of insulin.

Answer 1

It is given that .

a) The probability that, on a single test, x < 40 is

b) We have . Now, the sample mean .

So the correct choice is

The probability distribution of x is approximately normal with μ_x = 52 and σ_x = 17.68.

c) For , .

d) For , .

e) Yes. The probabilities decrease as n increases.