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 = 1.9% and standard deviation = 0.7%. (a) The fund has over 250 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 250 stocks in the fund. Why would this indicate that x has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 6.2.
The random variable (x-bar or x) is a mean of a sample size n = 250. By the (theory of normality, law of large numbers, or central limit theorem), the (x-bar or x) distribution is approximately normal.

(b) After 6 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? Hint: See Theorem 6.1, and assume that x has a normal distribution as based on part (a). (Round your answer to four decimal places.)

(c) After 2 years, what is the probability that 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?
Yes
No

Why would this happen?
The standard deviation (increases, decreases, or stays the same) as the (mean, sample size, average, or distribution) increases.

(e) If after 2 years the average monthly percentage return was less than 1%, would that tend to shake your confidence in the statement that = 1.9%? Might you suspect that has slipped below 1.9%? Explain.
a. This is very unlikely if = 1.9%. One would suspect that has slipped below 1.9%.
b. This is very likely if = 1.9%. One would suspect that has slipped below 1.9%.
c. This is very likely if = 1.9%. One would not suspect that has slipped below 1.9%.
d. This is very unlikely if = 1.9%. One would not suspect that has slipped below 1.9%.