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 = 57 and estimated standard deviation σ = 20. 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 μx = 57 and σx = 10.00. The probability distribution of x is approximately normal with μx = 57 and σx = 14.14. The probability distribution of x is approximately normal with μx = 57 and σx = 20. 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 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 lack of insulin. The more tests a patient completes, the weaker is the evidence for excess insulin

Answer 1

a) P(X < 40)

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

= P(Z < (40 - 57)/20)

= P(Z < -0.85)

= 0.1977

b) n = 2

= 57

      

The probability distribution of x is approximately normal with = 57 and = 14.14.

P( < 40)

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

= P(Z < (40 - 57)/14.14)

= P(Z < -1.202)

= 0.1147
c) n = 3

= 57

      

The probability distribution of x is approximately normal with = 57 and = 11.55.

P( < 40)

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

= P(Z < (40 - 57)/11.55)

= P(Z < -1.472)

= 0.0705

d) n = 5

= 57

      

The probability distribution of x is approximately normal with = 57 and = 8.94

P( < 40)

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

= P(Z < (40 - 57)/8.94)

= P(Z < -1.902)

= 0.0286

e) Yes, the probabilities decreased as n increased.

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