A certain mutual fund invests in both U.S. and foreign markets. Letxbe a random variable that represents the monthly percentage return for the fund. Assumexhas mean=1.9%and standard deviation=0.4%.(a) The fund has over250stocks that combine together to give the overall monthly percentage returnx. 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 returnxfor the fund is itself an average return computed using all250stocks in the fund. Why would this indicate thatxhas an approximately normal distribution? Explain.Hint:See the discussion after Theorem 6.2.

The random variable

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x

x-bar

is a mean of a sample sizen= 250. By the

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theory of normality

central limit theorem

law of large numbers

, the

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x-bar

x

distribution is approximately normal.

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

(c) After 2 years, what is the probability thatxwill 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 asn(number of months) increased?

Yes

No

Why would this happen?

The standard deviation

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increases

decreases

stays the same

as the

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mean

sample size

average

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 thathas slipped below1.9%? Explain.

This is very unlikely if= 1.9%. One would not suspect thathas slipped below 1.9%.

This is very likely if= 1.9%. One would not suspect thathas slipped below 1.9%.

This is very likely if= 1.9%. One would suspect thathas slipped below 1.9%.

This is very unlikely if= 1.9%. One would suspect thathas slipped below 1.9%.