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 = 52 and estimated standard deviation σ = 10. 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 μ_{x} = 52 and σ_{x} = 10.The probability distribution of x is approximately normal with μ_{x} = 52 and σ_{x} = 5.00. The probability distribution of x is approximately normal with μ_{x} = 52 and σ_{x} = 7.07.The probability distribution of x is not normal.
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?
YesNo
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 = = 52
standard deviation = = 10
a) P(x < 40) = P[(x - ) / < (40 - 52) / 10]
= P(z < -1.20)
Using z table,
= 0.1151
b) n = 2
_{} = = 52
_{} = / n = 10/ 2 = 7.07
The probability distribution of x is approximately normal with μx = 52 and σx = 7.07
P( < 40) = P(( - _{} ) / _{} < (40 - 52) / 7.07)
= P(z < -1.70)
Using z table
= 0.0446
c) n = 3
_{} = = 52
_{} = / n = 10/ 3 = 5.77
The probability distribution of x is approximately normal with μx = 52 and σx = 5.77.
P( < 40) = P(( - _{} ) / _{} < (40 - 52) / 5.77)
= P(z < -2.08)
Using z table
= 0.0188
d) n = 5
_{} = = 52
_{} = / n = 10/ 5 = 4.47
The probability distribution of x is approximately normal with μx = 52 and σx = 4.47
P( < 40) = P(( - _{} ) / _{} < (40 - 52) / 4.47)
= P(z < -2.68)
Using z table
= 0.0037
e) yes,
The more tests a patient completes, the weaker is the evidence for excess insulin