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 = 54 and estimated standard deviation σ = 47. 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 = 54 and σ_x = 33.23.The probability distribution of x is approximately normal with μ_x = 54 and σ_x = 23.50.    The probability distribution of x is not normal.The probability distribution of x is approximately normal with μ_x = 54 and σ_x = 47

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 weaker is the evidence for lack of insulin.    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.

Answer 1

a) P(X < 40)

= P((X - )/ < (40 - )/)

= P(Z < (40 - 54)/47)

= P(Z < -0.30)

= 0.3821

b) For n = 2

= 54

=

     = 47/ = 33.23

The probability distribution of is approximately normal with = 54 and = 33.23

P( < 40)

= P(( - )/( ) < (40 - )/( ))

= P(Z < (40 - 54)/(47/))

= P(Z < -0.42)

= 0.3372

c)

P( < 40)

= P(( - )/( ) < (40 - )/( ))

= P(Z < (40 - 54)/(47/))

= P(Z < -0.52)

= 0.3015

d) n = 5

P( < 40)

= P(( - )/( ) < (40 - )/( ))

= P(Z < (40 - 54)/(47/))

= P(Z < -0.67)

= 0.2514

e) Yes, the probabilities decreased as n increased.

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