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 = 50 and estimated standard deviation σ = 12. 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 = 50 and σ_x = 8.49.     The probability distribution of x is approximately normal with μ_x = 50 and σ_x = 12. The probability distribution of x is approximately normal with μ_x = 50 and σ_x = 6.00.

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

Answer 1

Solution,

Given that ,

mean = = 50

standard deviation = = 12

a) P(x < 40 ) = P[(x - ) / < ( 40 - 50 ) / 12 ]

= P(z < - 0.83 )

Using z table

= 0.2033

b) n = 2

=   = 50

= / n = 12 / 2 = 8.49

The probability distribution of x is approximately normal with = 50 and = 8.49

P( < 40 ) = P(( - ) / < ( 40 - 50 ) / 8.49 )

= P(z < -1.18 )

Using z table

= 0.1190

c) n = 3

= 50

= / n = 12 / 3 = 6.93

P( < 40) = P(( - ) / < ( 40 - 50 ) / 6.93 )

= P(z < -1.44 )

Using z table

= 0.0749

d ) n = 5

= 50

= / n = 12 / 5 = 5.37

P( < 40 ) = P(( - ) / < ( 40 - 50 ) / 5.37 )

= P(z < - 1.86 )

Using z table

= 0.0314

e) Yes,

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