Let x be a random variable that represents the level of glucose in the blood (mi

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, xhas a distribution that is approximately normal, with mean = 53 and estimated standard deviation = 27. 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 not normal.The probability distribution of x is approximately normal with x = 53 and x = 13.50.     The probability distribution of x is approximately normal with x = 53 and x = 27.The probability distribution of x is approximately normal with x = 53 and x = 19.09.

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

Explanation / Answer

a) Here x has a distribution that is approximately normal, with mean = 53 and estimated standard deviation = 27
So the probability that x < 40 is
= P(x < 40)
= P(z < ((40-53)/27))
= P(z < -0.48)
= 0.3156

b) It should seem reasonable that if several test are made, that their average would deviate less that of a single test. For a random sample of n individuals the sample standard deviation (s) will be less than the population standard deviation ().
s = /2

For n = 2 , s = 27/2 = 19.09

The probability distribution of x is approximately normal with x = 53 and x = 19.09.

z = (40 - 53)/19.09

z = -0.68

P = 0.2483

c)

For n = 3 , s = 27/3 = 15.59

The probability distribution of x is approximately normal with x = 53 and x = 15.59

z = (40 - 53)/15.59

z = -0.83

P = 0.2033

d)

For n = 5 , s = 27/5 = 15.59

The probability distribution of x is approximately normal with x = 53 and x = 12.07

z = (40 - 53)/12.07

z = -1.08

P = 0.1401

e) Yes, probabilities decrease as n increased

f) The more tests a patient completes, the weaker is the evidence for excess insulin.

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