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 = 71 and estimated standard deviation σ = 50. 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 = 71 and σx = 25.00. The probability distribution of x is approximately normal with μx = 71 and σx = 50. The probability distribution of x is approximately normal with μx = 71 and σx = 35.36. 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 stronger is the evidence for lack of insulin. The more tests a patient completes, the weaker is the evidence for excess insulin.

Answer 1

Solution :

Given that ,

mean = = 71

standard deviation = = 50

a) P(x < 40) = P[(x - ) / < (40 - 71) / 50]

= P(z < -0.62)

Using z table,

= 0.2676

b) n = 2

= = 71

= / n = 50/ 2 = 35.36

The probability distribution of x is approximately normal with μx = 71 and σx = 35.36.

P( < 40) = P(( - ) / < (40 - 71) / 35.36)

= P(z < -0.88)

Using z table

= 0.1894

c) n = 3

= = 71

= / n = 50/ 3 = 28.87

The probability distribution of x is approximately normal with μx = 71 and σx = 28.87

P( < 40) = P(( - ) / < (40 - 71) / 28.87)

= P(z < -1.07)

Using z table

= 0.1423

d) n = 5

= = 71

= / n = 50/ 5 = 22.36

The probability distribution of x is approximately normal with μx = 71 and σx = 22.36

P( < 40) = P(( - ) / < (40 - 71) / 22.36)

= P(z < -1.39)

Using z table

= 0.0823

e) yes,

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