Question

Consider a population of 10241024 mutual funds that primarily invest in large companies. You have determined that muμ​, the mean​ one-year total percentage return achieved by all the​ funds, is 8.408.40 and that sigmaσ​,the standard​ deviation, is 3.503.50. Complete​ (a) through​ (c). a. According to the empirical​ rule, what percentage of these funds is expected to be within ​±33 standard deviations ,deviations of the​ mean? 99.799.7​% b. According to the Chebyshev​ rule, what percentage of these funds are expected to be within

​±22 standard deviations of the​ mean? -75.075.0​% ​(Round to two decimal places as​ needed.)

***** c. According to the Chebyshev​ rule, at least

88.8988.89​%

of these funds are expected to have​ one-year total returns between what two​ amounts?

Between_ and _.

Answer 1

We have from the data        
μ = Mean = 8.40       
σ = SD = 3.50       
n = 1024       
        
Empirical Rule :       
The empirical rule states that for a normal distribution,        
nearly all of the data will fall within three standard deviations of the mean.        
The empirical rule can be broken down into three parts:       
68% of data falls within the first standard deviation from the mean.       
95% fall within two standard deviations.       
99.7% fall within three standard deviations.       
        
a) Percentage of these funds is expected to be within ​±3 standard deviations of the​ mean       
is 99.7%       
Answer : 99.70%       
        
b) "Chebyshev’s inequality states that for any probability distribution of a random variable
X, "       
and any number k, greater than 1, the probability that the value of X lies at least k       
standard deviations from its mean is at most 1/k2 for each given k.       

To find percentage of the funds expected to be between ±2 standard deviations of the mean.       
So Chebyshev’s inequality says that at least 75% of the data values of any distribution        
must be within 2 standard deviations of the mean.       
Answer : 75.00%       
        
c) To find k in Chebyshev's inequality such that        

Hence, k = 3       
kσ = 3*3.5 = 10.5       

that is, -2.1 < X < 18.9       

88.89​% of these funds are expected to have​ one-year total returns between -2.1 and 18.9