Medical: 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 = 85 and estimated standard deviation = 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 7.1. 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 theprobabilities 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.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Medical: 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 = 85 and estimated standard deviation = 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 7.1. 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.
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