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 = 84 and estimated standard deviation σ = 29. 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 = 84 and σ_x = 29.    The probability distribution of x is approximately normal with μ_x = 84 and σ_x = 20.51.The probability distribution of x is approximately normal with μ_x = 84 and σ_x = 14.50.

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 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 excess insulin.

Answer 1

a) X represents the glucose level       
X follows normal distribution with μ = 84 and σ = 29       
To find P(X < 40)       
We use Excel function NORM.DIST to find the probability       
P(X < 40) = NORM.DIST(40, 84, 29, TRUE)       
                   = 0.0646       
P(X < 40) = 0.0646       
        
b) X represents the average glucose level       
for n = 2       
By Central Limit Theorem, the correct answer is       
The probability distribution of x is approximately normal with μx = 84 and σx = σ/√n       
σx = 29/√2   = 20.5061 = 20.51       
Answer :       
The probability distribution of x is approximately normal with μx = 84 and σx = 20.51       
P(X < 40) = NORM.DIST(40, 84, 20.51, TRUE)       
                   = 0.01597       
P(X < 40) = 0.0646       
        
c) X represents the average glucose level       
for n = 3       
By Central Limit Theorem, the correct answer is       
The probability distribution of x is approximately normal with μx = 84 and σx = σ/√n       
σx = 29/√3   = 16.74       
Answer :       
The probability distribution of x is approximately normal with μx = 84 and σx = 16.74       
P(X < 40) = NORM.DIST(40, 84, 16.74, TRUE)       
                   = 0.0043       
P(X < 40) = 0.0043       
        
d) X represents the average glucose level       
for n = 5       
By Central Limit Theorem, the correct answer is       
The probability distribution of x is approximately normal with μx = 84 and σx = σ/√n       
σx = 29/√5   = 12.97       
Answer :       
The probability distribution of x is approximately normal with μx = 84 and σx = 12.97       
P(X < 40) = NORM.DIST(40, 84, 12.97, TRUE)       
                   = 0.00035       
P(X < 40) = 0.00035       
        
e) We have following probabilities for different n       

n	Probability
2	0.0646
3	0.0043
5	0.00035

We can see that Probability decreases as n increases       
Answer :       
YES       
        
f) This implies       
The more tests a patient completes, the weaker is the evidence for excess insulin.