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 = 30. 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 7.1.

The probability distribution of x is not normal.The probability distribution of x is approximately normal with μ_x = 71 and σ_x = 15.00.    The probability distribution of x is approximately normal with μ_x = 71 and σ_x = 21.21.The probability distribution of x is approximately normal with μ_x = 71 and σ_x = 30.

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 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)
mean = 71 , sigam = 340

P(x< 40)
= P(z< (x -mean)/sigma)
= P(z< (40 -71)/30)
= P(z< -1.03)
= 0.1507

b)

n = 2

mean = 71 , std.dev = sigma/sqrt(n)
= 30/sqrt(2) = 21.21

The probability distribution of x is approximately normal with μx = 71 and σx = 21.21

P(x< 40)
= P(z< (x -mean)/sigma)
= P(z< (40 -71)/21.21)
= P(z< -1.46)
= 0.0720

c)

n = 3

mean = 71 , std.dev = sigma/sqrt(n)
= 30/sqrt(3) = 17.32

The probability distribution of x is approximately normal with μx = 71 and σx = 17.32

P(x< 40)
= P(z< (x -mean)/sigma)
= P(z< (40 -71)/17.32)
= P(z< -1.79)
= 0.0367

d)

n = 5

mean = 71 , std.dev = sigma/sqrt(n)
= 30/sqrt(5) = 13.42

The probability distribution of x is approximately normal with μx = 71 and σx = 13.42

P(x< 40)
= P(z< (x -mean)/sigma)
= P(z< (40 -71)/13.42)
= P(z< -2.31)
= 0.0104

e)

yes, as the sample size increased probabilities decreased

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