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 = 95 and estimated standard deviation σ = 39. 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 μ_x = 95 and σ_x = 27.58.The probability distribution of x is approximately normal with μ_x = 95 and σ_x = 19.50.    The probability distribution of x is approximately normal with μ_x = 95 and σ_x = 39.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?

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 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.The more tests a patient completes, the weaker is the evidence for lack of insulin.

Answer 1

Let X represent level of glucose in blood.

X follows Normal distribution with mean = 95 and standard deviation =39.

According to the properties of Normal distribution, follows standard normal distribution.

Part a) -

P[ X<40 ] = = P[ Z< -1.41 ] = 1 - P[ Z> -1.41] = 1 - P[ Z< 1.41] = 1-0.9207 = 0.0793

Part b) - Let denote the average for two tests taken about a week apart.

According to the properties of Normal distribution follows Normal distribution with same mean and standard deviation

Here, n=2

Hence, ~ N( , )

Hence, follows Normal distribution with mean=95 and standard deviation= 27.58

follows Standard Normal distribution.

P[ <40 ] = = P[ Z< -1.99] = 1 - P[ Z< 1.99] = 1- 0.9767 = 0.0233

Part c) -

Let denote the average for three tests.

Here, n=2

Hence, ~ N( , )

follows Standard Normal distribution.

P[ <40 ] = = P[ Z< -2.44] = 1 - P[ Z< 2.44] = 1- 0.9927 = 0.0073

Part d) -

Let denote the average for five tests.

Here, n=5

Hence, ~ N( , )

follows Standard Normal distribution.

P[ <40 ] = = P[ Z< -3.15] = 1 - P[ Z< 3.15] = 1- 0.9992 = 0.0008

Part e) -

Compairing the results from parts a, b, c, and d we can see that the probabilities for X<40 decreased.

Glucose level being less than the mean value 95 means low glucose level.

We know that low level of glucose implies there is excess insulin in the body.

Hence, this implies that the more tests a patient completes, the weaker is the evidence for excess insulin.

I hope you find the solution helpful. If you have any doubt then feel free to ask in the comment section.

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