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 = 90and estimated standard deviation σ = 49. 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 μ_x = 90 and σ_x = 24.50.    The probability distribution of x is approximately normal with μ_x = 90 and σ_x = 49.The probability distribution of x is approximately normal with μ_x = 90 and σ_x = 34.65.

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?

Yes

No   

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 lack of insulin.

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.

Answer 1

a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	90
std deviation   =σ=	49.0000

probability =

P(X<40)

=

P(Z<-1.02)=

0.1539

b)

The probability distribution of x is approximately normal with μ_x = 90 andσ_x = 34.65.

probability =

P(X<40)

=

P(Z<-1.44)=

0.0749

c)

probability =

P(X<40)

=

P(Z<-1.77)=

0.0384

d)

probability =

P(X<40)

=

P(Z<-2.28)=

0.0113

e)

Yes

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