Question

The quarterly returns for a group of 6161 mutual funds are well modeled by a Normal model with a mean of 6.16.1​% and a standard deviation of 2.72.7​%. Use the​ 68-95-99.7 Rule to find the cutoff values that would separate the following percentages of​ funds, rather than using technology to find the exact values. ​a) the highest 5050​% ​b) the highest 1616​% ​c) the lowest 2.52.5​% ​d) the middle 6868​%

Answer 1

Answer:

a) The cut off value of the highest 50% is 6.16%.

b) The cut off value of the highest 16% is 8.8%.

c) The cut off value of the lowest 2.5% is 0.7%.

d) The cut off values of the middle 68% is 3.4% for the lower bound and 8.8% for the higher bound.

Step-by-step explanation:

The 68–95–99.7 rule is an empirical rule to remember the percentage of values that lie within a band around the mean in a normal distribution with a width of two, four and six standard deviations respectively.

a) Cut off value of the highest 50%

It corresponds to the half that has the higher returns, which are the ones that are above the mean (0 standard deviation). The cut off value in this case is the mean x=6.16%.

b) Cut off value of the highest 16%

In this case, we use the 68 rule-value, which means that 68% of the values lies within a band around the mean with a width of two standard deviations.

The percentage that lies out of this band is 100%-68%=32%, of which 16% lies above the band (the highest 16%) and 16% lies below the band.

We can estimate, according to this rule, that the highest 16% are above the cutoff value X of μ+1σ:

$X=\mu+\sigma=6.1%+2.7%=8.8%$

The cut off value of the highest 16% is 8.8%.

c) Cut off of the lowest 2.5%

In this case, we use the 95 rule.

The lowest 2.5% lies below two standards deviation less of the mean

$X=\mu-2*\sigma=6.1%-2*2.7%=0.7%$

The cut off value of the lowest 2.5% is 0.7%.

d) Cut off of the middle 68%

In this case we calculate the band within one standard deviation.

$X_{low}=\mu-\sigma=6.1%-2.7%=3.4%\\\\X_{high}=\mu+\sigma=6.1%+2.7%=8.8%$

The cut off values of the middle 68% is 3.4% for the lower bound and 8.8% for the higher bound.