In: Statistics and Probability

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 = 51 and estimated standard deviation σ = 47. 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 three 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? The probability distribution of x is approximately normal with μx = 51 and σx = 23.50. The probability distribution of x is approximately normal with μx = 51 and σx = 47. The probability distribution of x is approximately normal with μx = 51 and σx = 33.23. The probability distribution of x is not normal. What is the probability that x < 40? (Round your answer to three decimal places.) (c) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to three decimal places.) (d) Repeat part (b) for n = 5 tests taken a week apart. (Round your answer to three 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 weaker is the evidence for lack of insulin. 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.

Solution :

Given that ,

mean = = 51

standard deviation = = 47

a) P(x < 40) = P[(x - ) / < (40 - 51) / 47]

= P(z < -0.23)

Using z table,

= 0.4090

b) n = 2

= = 51

= / n = 47/ 2 = 33.23

The probability distribution of x is approximately normal with μx = 51 and σx = 33.23

P( < 40) = P(( - ) / < (40 - 51) / 33.23)

= P(z < -0.33)

Using z table

= 0.3707

c) n = 3

= = 51

= / n = 47/ 3 = 27.14

The probability distribution of x is approximately normal with μx = 51 and σx = 27.14

P( < 40) = P(( - ) / < (40 - 51) / 27.14)

= P(z < -0.41)

Using z table

= 0.3409

d) n = 5

= = 51

= / n = 47/ 5 = 21.02

The probability distribution of x is approximately normal with μx = 51 and σx = 21.02

P( < 40) = P(( - ) / < (40 - 51) / 21.02)

= P(z < -0.52)

Using z table

= 0.3015

e) yes,

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

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