Question

5. Blood Glucose 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 =85 and estimated standard deviation ợ =25 (based on information from Diagnostic Tests with Nursing Applications, edited by S. Loeb, Springhouse). 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? (b) Suppose a doctor uses the average �̅for two tests taken about a week apart. What can we say about the probability distribution of�̅? Hint: See Theorem 6.1. What is the probability that �̅< 40? (c) Repeat part (b) for n = 3 tests taken a week apart. (d) Repeat part (b) for n = 5 tests taken a week apart. (e) Interpretation Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased? Explain what this might imply if you were a doctor or a nurse. If a patient had a test result of �̅< 40 based on five tests, explain why either you are looking at an extremely rare event or (more likely) the person has a case of excess insulin.

Answer 1

Given and .

a)

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b) Given

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c) Given

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d) Given

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e) Yes, the probabilities decreased as n increased. If a patient had a test result of x¯< 40 based on five tests, this means severe excess insulin, and medication should be prescribed. from part d, it is highly unusual to get average of 5 tests less than 40, if actually level of glucose in the blood (milligrams per deciliter of blood) after a 12-hour fast is 85.