Question

Please explain how you compute these answers given for A,B,C. 

Objective:Apply the empirical rule and the Chebyshev rule. 

Consider a population of 1024 mutual funds that primarily invest in large companies You have determined that p, the mean one-year total percentage return achieved by all the funds, is 9.20 and that o, the standard deviation, is 1 25. 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? 

99.7 % 

b. According to the Chebyshev rule, what percentage of these funds are expected to be within ±2 standard deviations of the mean? 

75 % (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? 

Between 5.45 and 12.95   (Round to two decimal places as needed.) 

Answer 1

a) According to the empirical rule, 99.7 % of the observations lies within 3 standard deviations from the mean. Therefore 99.7% is correct here.

b) According to Chebyshev's inequality theorem, at least 1 - 1/k² of the observations lies within k standard deviations from the mean. Therefore at least 1 - 1/2² = 75% of the observations lies within 2 standard deviations from the mean. Therefore 75% is correct here.

c) Now here we are given that at least 88.89% of the observations lies within what interval.

88.89% is 8/9

1 - 1/k² = 8/9

1/k² = 1 - 8/9

k = 3 here.

Therefore we just need to find the interval as:

Mean - 3*Std Dev and Mean + 3*Std Dev

9.2 - 3*1.25 = 5.45 and 9.2 + 3*1.25 = 12.95

Therefore they lie between 5.45 and 12.95