Let x be a random variable that represents the level ofglucose in the blood (milligrams per deciliter of blood) after a12-hour fast. Assume that for people under 50 years old, x has adistribution that is approximately normal, with mean μ = 64 andestimated standard deviation σ = 40. A test result x < 40 is anindication of severe excess insulin, and medication is usuallyprescribed.
(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 takenabout a week apart. What can we say about the probabilitydistribution of x? (multiple choice options below)
The probability distribution of x is approximately normal withμx = 64 and σx = 28.28.
The probability distribution of x is approximately normal withμx = 64 and σx = 20.00.
The probability distribution of x is approximately normal withμx = 64 and σx = 40.
The probability distribution of x is not normal.
(b2) What is the probability that x < 40? (Round youranswer to four decimal places.)
(c) Repeat part (b) for n = 3 tests taken a week apart. (Roundyour answer to four decimal places.)
(d) Repeat part (b) for n = 5 tests taken a week apart. (Roundyour answer to four decimal places.)
(e) Compare your answers to parts (a), (b), (c), and (d). Didthe probabilities decrease as n increased?
yes
no
(e2) Explain what this might imply if you were a doctor or anurse.
The more tests a patient completes, the stronger is theevidence for excess insulin.
The more tests a patient completes, the weaker is the evidencefor lack of insulin.
The more tests a patient completes, the stronger is theevidence for lack of insulin.
The more tests a patient completes, the weaker is the evidencefor excess insulin.
(f)A certain mutual fund invests in both U.S. and foreign markets. Letx be a random variable that represents the monthly percentagereturn for the fund. Assume x has mean μ = 1.8% and standarddeviation σ = 0.6%.
(a) The fund has over 275 stocks that combine together to givethe overall monthly percentage return x. We can consider themonthly return of the stocks in the fund to be a sample from thepopulation of monthly returns of all world stocks. Then we see thatthe overall monthly return x for the fund is itself an averagereturn computed using all 275 stocks in the fund. Why would thisindicate that x has an approximately normal distribution? Explain.Hint: See the discussion after Theorem 7.2.
The random variable is a mean of a sample size n = 275. By the, the distribution is approximately normal.
(g) After 6 months, what is the probability that the averagemonthly percentage return x will be between 1% and 2%? Hint: SeeTheorem 7.1, and assume that x has a normal distribution as basedon part (a). (Round your answer to four decimal places.)
(h) After 2 years, what is the probability that x will bebetween 1% and 2%? (Round your answer to four decimalplaces.)
(i) Compare your answers to parts (b) and (c). Did theprobability increase as n (number of months) increased?
(j) If after 2 years the average monthly percentage return wasless than 1%, would that tend to shake your confidence in thestatement that μ = 1.8%? Might you suspect that μ has slipped below1.8%? (multiple choice)
This is very likely if μ = 1.8%. One would not suspect that μhas slipped below 1.8%.
This is very unlikely if μ = 1.8%. One would not suspect thatμ has slipped below 1.8%.
This is very unlikely if μ = 1.8%. One would suspect that μhas slipped below 1.8%.
This is very likely if μ = 1.8%. One would suspect that μ hasslipped below 1.8%.
