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 = 57 and estimated standard deviation σ = 38. 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.) .3264 Correct: Your answer is correct.

(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 7.1. The probability distribution of x is not normal. The probability distribution of x is approximately normal with μx = 57 and σx = 19.00. The probability distribution of x is approximately normal with μx = 57 and σx = 38. The probability distribution of x is approximately normal with μx = 57 and σx = 26.87. Correct: Your answer is correct.

What is the probability that x < 40? (Round your answer to four decimal places.) .2451 Incorrect: Your answer is incorrect.

(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? Yes No

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

Answer 1

(a)

P(X < 40) = P[Z < (40 - 57)/38] = P[Z < -0.45] = 0.3264 (Using Z tables)

(b)

By Central limit theorem, the probability distribution of x is approximately normal with μx = 57 and σx =38/ = 26.87

The probability distribution of x is approximately normal with μx = 57 and σx = 26.87.

P(X < 40) = P[Z < (40 - 57)/26.87] = P[Z < -0.63] = 0.2643 (Using Z tables)

(c)

By Central limit theorem, the probability distribution of x is approximately normal with μx = 57 and σx =38/ = 21.94

The probability distribution of x is approximately normal with μx = 57 and σx = 21.94.

P(X < 40) = P[Z < (40 - 57)/21.94] = P[Z < -0.77] = 0.2206 (Using Z tables)

(d)

By Central limit theorem, the probability distribution of x is approximately normal with μx = 57 and σx =38/ = 16.99

The probability distribution of x is approximately normal with μx = 57 and σx = 16.99.

P(X < 40) = P[Z < (40 - 57)/16.99] = P[Z < -1.00] = 0.1587 (Using Z tables)

(e)

Yes, the probabilities decrease as n increased.

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