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, x has a distribution that is approximately normal, with mean = 63 and estimated standard deviation = 31. 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 = 63 and x = 21.92.    The probability distribution of x is approximately normal with x = 63 and x = 15.50.The probability distribution of x is approximately normal with x = 63 and x = 31.

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?

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

Explanation / Answer

a)

probability that, on a single test, x < 40 =P(X<40)=P(Z<(40-63)/31)=P(Z<-0.7419)=0.2291

b)

here from central limit theorum ; mean of samples =63

and std error of mean =std deviation/(n)1/2 =31/(2)1/2 =21.92

hence option The probability distribution of x is approximately normal with x = 63 and x = 21.92 is correct

P(Xbar<40)=P(Z<(40-63)/21.92)=P(Z<-1.0493)=0.1470

c)

std error of mean =std deviation/(n)1/2 =31/(3)1/2 =17.90

P(Xbar<40)=P(Z<(40-63)/17.90)=P(Z<-1.2851)=0.0994

d)

std error of mean =std deviation/(n)1/2 =31/(5)1/2 =13.86

P(Xbar<40)=P(Z<(40-63)/13.86)=P(Z<-1.6590)=0.0486

e)

yes ,probabilities decrease as n increased

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

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