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 = 79 and estimated standard deviation σ = 32. 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 = 79 and σx = 22.63. The probability distribution of x is not normal. The probability distribution of x is approximately normal with μx = 79 and σx = 16.00. The probability distribution of x is approximately normal with μx = 79 and σx = 32. 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? Yes No Explain what this might imply if you were a doctor or a nurse. The more tests a patient completes, the weaker is the evidence for excess insulin. The more tests a patient completes, the stronger 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 lack of insulin. Need Help? Read It Watch It

Answer 1

Given Normal distribution N(79,32)

mean μ = 79 and estimated standard deviation σ = 32

a) the probability that, on a single test, x < 40

P(X<40)

for x =40, Z = (40 - 79)/32 = -1.21875

P(X<40) = P(Z<-1.21875) = 0.1115

b) The sample follows approximately normal distribution N(μ , )

The probability distribution of x is approximately normal with μx = 79 and σx = 22.63.

the probability that x < 40

= -1.7236

P(X<40) = P(Z<-1.7236) = 0.0424

c) for n = 3

Z = -2.1109

P(X<40) = P(Z<-2.1109) = 0.0174

d) for n = 5

Z = -2.7252

P(X<40) = P(Z<-2.7252) =0.0032

e) as the n increases the probability decreases, The more tests a patient completes, the weaker is the evidence for excess insulin.