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 = 56 and estimated standard deviation σ = 24. 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 μx = 56 and σx = 16.97. The probability distribution of x is approximately normal with μx = 56 and σx = 24. The probability distribution of x is approximately normal with μx = 56 and σx = 12.00. The probability distribution of x is not normal. 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 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.

Answer 1

Given that ,

mean = = 56

standard deviation = = 24

a) P(x < 40) = P[(x - ) / < ( 40 - 56) / 24 ]

= P(z < -0.67 )

Using z table

= 0.2514

b) n = 2

= 56

= / n = 24 / 2 = 16.97

The probability distribution of x is approximately normal with = 56 and = 16.97

P( < 40) = P(( - ) / < ( 40 - 56) / 16.97)

= P(z < -0.94 )

Using z table

= 0.1736

c) n = 3

= 56

= / n = 24 / 3 = 13.86

P( < 40 ) = P(( - ) / < ( 40 - 56) / 13.86)

= P(z < -1.15 )

Using z table

= 0.1251

d) n = 5

= 56

= / n = 24 / 5 = 10.73

P( < 40 ) = P(( - ) / < ( 40 - 56) / 10.73 )

= P(z < -1.49 )

Using z table

= 0.0681

e) Yes,the probabilities decrease as n increased.

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