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 ? = 59 and estimated standard deviation ? = 45. 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 = 59 and ?_x = 45.    The probability distribution of x is approximately normal with ?_x = 59 and ?_x = 22.50.The probability distribution of x is approximately normal with ?_x = 59 and ?_x = 31.82.

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

Answer 1

a)

P(X<40)=P(Z<(40-59)/45)=P(Z<-0.42)=0.3372 ( please try 0.3364 if this comes wrong)

b).The probability distribution of x is approximately normal with x = 59 and x = 31.82.

P(Xbar<40)=P(Z<(40-59)/(45/sqrt(2))=P(Z<-0.60)=0.2743 (please try 0.2752 if this comes wrong)

c)

P(Xbar<40)=P(Z<(40-59)/(45/sqrt(3))=P(Z<-0.73)=0.2327 (please try 0.2323 if this comes wrong)

d)

P(Xbar<40)=P(Z<(40-59)/(45/sqrt(5))=P(Z<-0.94)=0.1736 (please try 0.1726 if this comes wrong)

e)

yes

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