Let x be a random variable that represents the level of glucose in the blood (mi

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 = 63 and estimated standard deviation = 28. 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 x = 63 and x = 28.The probability distribution of x is approximately normal with x = 63 and x = 19.80.    The probability distribution of x is approximately normal with x = 63 and x = 14.00.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 stronger is the evidence for lack of 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.The more tests a patient completes, the weaker is the evidence for excess insulin.

Explanation / Answer

(a) What is the probability that, on a single test, x < 40? (Round your answer to four decimal places)

Ans: x<40; = 63 ; = 28 ; n=1 (Single test is conducted)

Z-calc. = (x-)/(/n)

Imputing values

Z-calc. = (40 - 63)/28 = -0.8214

Checking the z distribution table for area under the curve.

Since we are talking about x<40, hence the cummulative value from 40 to left hand of the curve to be considered.

Prob. (x<40) = .2057

b) If Two tests are conducted

Ans: x<40; = 63 ; = 28 ; n=2 (Two test is conducted)

In that case, Std. Dev will become /sqrt(n) i.e. 28/sqrt(2) = 19.80

Hence for 2 tests ; The probability distribution of x is approximately normal with x = 63 and x = 19.80

Z calc. = (40-63)/19.80 = -1.161

Since we are talking about x<40, hence the cummulative value from 40 to left hand of the curve to be considered.

Prob. (x<40) = .1227

c) If Three tests are conducted

Ans: x<40; = 63 ; = 28 ; n=3 (Three test is conducted)

In that case, Std. Dev will become /sqrt(n) i.e. 28/sqrt(3) = 16.17

Hence for 3 tests ; The probability distribution of x is approximately normal with x = 63 and x = 16.17

Z calc. = (40-63)/16.17 = -1.423

Since we are talking about x<40, hence the cummulative value from 40 to left hand of the curve to be considered.

Prob. (x<40) = .0774

d) If five tests are conducted

Ans: x<40; = 63 ; = 28 ; n=5 (Five test is conducted)

In that case, Std. Dev will become /sqrt(n) i.e. 28/sqrt(5) = 12.52

Hence for 2 tests ; The probability distribution of x is approximately normal with x = 63 and x = 12.52

Z calc. = (40-63)/16.17 = -1.837

Since we are talking about x<40, hence the cummulative value from 40 to left hand of the curve to be considered.

Prob. (x<40) = .0331

e) Yes, probability is dec. as n increases

Inferences :

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

As the probability of getting an excess insulin level i.e. x<40 is decreasing as the sample size is increased.

Hence the awnser!!

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.