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 = 65 and estimated standard deviation σ = 31. 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 = 65 and σ_x = 21.92.    The probability distribution of x is approximately normal with μ_x = 65 and σ_x = 31.The probability distribution of x is approximately normal with μ_x = 65 and σ_x = 15.50.

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 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 stronger is the evidence for lack of insulin.

Answer 1

Solution :

Given that ,

mean = = 65

standard deviation = = 31

a) P(x < 40) = P[(x - ) / < (40 - 65) / 31]

= P(z < -0.81)

Using z table,

= 0.2090

b) n = 2

= = 65

= / n = 31/ 2 = 21.92

The probability distribution of x is approximately normal with μx = 65 and σx = 21.92.

P( < 40) = P(( - ) / < (40 - 65) / 21.92)

= P(z < -1.14)

Using z table

= 0.1271

c) n = 3

= = 65

= / n = 31/ 3 = 17.90

The probability distribution of x is approximately normal with μx = 65 and σx = 17.90.

P( < 40) = P(( - ) / < (40 - 65) / 17.90)

= P(z < -1.40)

Using z table

= 0.0808

d) n = 5

= = 65

= / n = 31/ 5 = 13.86

The probability distribution of x is approximately normal with μx = 65 and σx = 13.86

P( < 40) = P(( - ) / < (40 - 65) / 13.86)

= P(z < -1.80)

Using z table

= 0.0359

e) yes,

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