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 = 64 and estimated standard deviation σ = 40. 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? (multiple choice options below)

The probability distribution of x is approximately normal with μx = 64 and σx = 28.28.

The probability distribution of x is approximately normal with μx = 64 and σx = 20.00.

The probability distribution of x is approximately normal with μx = 64 and σx = 40.

The probability distribution of x is not normal.

(b2) 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

(e2) 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 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 weaker is the evidence for excess insulin.

(f) A certain mutual fund invests in both U.S. and foreign markets. Let x be a random variable that represents the monthly percentage return for the fund. Assume x has mean μ = 1.8% and standard deviation σ = 0.6%.

(a) The fund has over 275 stocks that combine together to give the overall monthly percentage return x. We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return x for the fund is itself an average return computed using all 275 stocks in the fund. Why would this indicate 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 average monthly percentage return x will be between 1% and 2%? Hint: See Theorem 7.1, and assume that x has a normal distribution as based on part (a). (Round your answer to four decimal places.)

(h) After 2 years, what is the probability that x will be between 1% and 2%? (Round your answer to four decimal places.)

(i) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased?

(j) If after 2 years the average monthly percentage return was less than 1%, would that tend to shake your confidence in the statement that μ = 1.8%? Might you suspect that μ has slipped below 1.8%? (multiple choice)

This is very likely if μ = 1.8%. One would not suspect that μ has slipped below 1.8%.

This is very unlikely if μ = 1.8%. One would not suspect that μ has slipped below 1.8%.

This is very unlikely if μ = 1.8%. One would suspect that μ has slipped below 1.8%.

This is very likely if μ = 1.8%. One would suspect that μ has slipped below 1.8%.

Answer 1