Question

# Let x be a random variable that represents the level of glucose in the blood (milligrams...

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 = 56 and estimated standard deviation σ = 42. 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 μx = 56 and σx = 21.00.The probability distribution of x is not normal.     The probability distribution of x is approximately normal with μx = 56 and σx = 29.70.The probability distribution of x is approximately normal with μx = 56 and σx = 42.

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.     T

he 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.

Solution:-

(a) Given that μ = 56 , σ = 42

P(X < 40) = P((X-μ)/σ < (40-56)/42)
= P(Z < -0.3810)
= 0.3520

(b) Given that μx = 56, σx = 21.00

P(X < 40) = P(Z < (40-56)/21)
= P(Z < -0.7619)
= 0.2236

(c) For n = 3, σx = 24.25

P(X < 40) = P(Z < 40-56)/24.25)
= P(Z < -0.6598)
= 0.2546

(d) For n = 5, σx = 18.78

P(X < 40) = P(Z < 40-56)/18.78)
= P(Z < -0.8520)
= 0.1977

(e) yes.

=>option B.The more tests a patient completes,the stronger is the evidence for excess insulin

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