Question

Consider a population of 1024 mutual funds that primarily invest in large companies. You have determined that μ​, the mean​ one-year total percentage return achieved by all the​ funds, is 9.70 and that σ​, the standard​ deviation, is 1.75. Complete​ (a) through​ (c).

a. According to the empirical​ rule, what percentage of these funds is expected to be within ±3 standard deviations of the​ mean?

b. According to the Chebyshev​ rule, what percentage of these funds are expected to be within ±2 standard deviations of the​ mean ​(Round to two decimal places as​ needed.)

c. According to the Chebyshev​ rule, at least 88.89% of these funds are expected to have​ one-year total returns between what two​ amounts? ​(Round to two decimal places as​ needed.)

Answer 1

(a)

The empirical rule states that 68.27%, 95.45% and 99.73% of the data lies within one standard deviation, two standard deviations and three standard deviations of the mean respectively.

Thus, we expect 99.73% of these funds to be within 3 standard deviations of the mean.

(b)

The chebychey's rule states that at least 1 - 1/k² of the values of a distribution are within k standard deviations of the mean.

Thus, 1 - 1/2² of the values are expected to be within 2 standard deviations of the mean

Thus, 75.00% of these funds are expected to be within 2 standard deviations of the mean, according to the Chebychev rule.

(c)

The chebychey's rule states that at least 1 - 1/k² of the values of a distribution are within k standard deviations of the mean.

Now, we are given that at least 88.89% of these funds are expected to have one-year total returns within, say k, standard deviations of the mean.

Now,

Thus, atleast 88.89% of these funds are expected to have one-year total returns within 3 standard deviations of the mean.

Thus, at least 88.89% of these funds are expected to have one-year total returns between

(9.7 - 3*1.75 , 9.7 + 3*1.75) = (4.45, 14.95).

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