Question

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 125 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean ? = 1.2% and standard deviation ? = 1.4%.

(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 125 stocks in the fund) has a distribution that is approximately normal? Explain.

---Select--- Yes No , x is a mean of a sample of n = 125 stocks. By the  ---Select--- law of large numbers theory of normality central limit theorem , the x distribution  ---Select--- is is not approximately normal.

(b) After 9 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)


(c) After 18 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)


(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the mean increases.Yes, probability increases as the standard deviation decreases.     Yes, probability increases as the standard deviation increases.No, the probability stayed the same.

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that ? = 1.2%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)
P(x > 2%) =  

Explain.

This is very unlikely if ? = 1.2%. One would suspect that the European stock market may be heating up.This is very unlikely if ? = 1.2%. One would not suspect that the European stock market may be heating up.     This is very likely if ? = 1.2%. One would not suspect that the European stock market may be heating up.This is very likely if ? = 1.2%. One would suspect that the European stock market may be heating up.

Answer 1

(a) Yes, it is reasonable to assume that x (the average monthly return on the 125 stocks in the fund) has a distribution that is approximately normal.  By the central limit theorem , the x distribution is approximately normal.

(b) Here standard error of mean stock return = standard deviation/sqrt(Time period) = 1.4/sqrt(9) = 0.4667

if is the average monthly percentage return

Pr(1% < < 2%) = NORM( < 2%; 1.2% ; 0.4667%) - NORM( < 1%; 1.2% ; 0.4667%)

Z₂ = (2 - 1.2)/0.4667 = 1.7142

Z₁ = (1 - 1.2)/0.4667 = -0.4285

Pr(1% < < 2%) = NORM( < 2%; 1.2% ; 0.4667%) - NORM( < 1%; 1.2% ; 0.4667%)

= Pr(Z< 1.7142) - Pr(Z < -0.1167)

= 0.9567 - 0.3341

= 0.6226

(c) Now time period = 18 months

standard error of sample mean = 1.4/sqrt(18) = 0.33%

Pr(1% < < 2%) = NORM( < 2%; 1.2% ; 0.33%) - NORM( < 1%; 1.2% ; 0.33%)

= Pr(Z< 2.4242) - Pr(Z < -0.6061)

= 0.9923 - 0.2722

= 0.7201

(d) Yes, probability increases as the standard deviation increases. Option D is correct.

(e) Here

Pr( > 2%) = NORM( > 2% ; 1.2% ; 0.3%) = 1 - NORM( < 2% ; 1.2% ; 0.33%)

Z = (2 - 1.2)/0.33 = 2.4242

Pr( > 2%) = NORM( > 2% ; 1.2% ; 0.3%) = Pr(Z > 2.4242) = 1 - 0.9923 = 0.0077

This is very unlikely if ? = 1.2%. One would suspect that the European stock market may be heating up.