Question

In the last quarter of​ 2007, a group of 64 mutual funds had a mean return of 5.9% with a standard deviation of 7.1​%.

If a normal model can be used to model​ them, what percent of the funds would you expect to be in each​ region? Use the​ 68-95-99.7 rule to approximate the probabilities rather than using technology to find the values more precisely. Be sure to draw a picture first.

​a) Returns of negative −1.2​% or less	​b) Returns of 5.9​% or more
​c) Returns between negative −8.3​% and 20.1​%	​d) Returns of more than 27.227.2​%

Answer 1

a) -1.2 = 5.9 - 7.1 that is 1 standard deviation less than the mean. Using the 68-95-99.7 rule of the normal distribution, we know that 68% of the observations lies within 1 standard deviation of the mean. This means that (1 - 0.68)/2 = 16% of the observations lies below -1.2%.

Therefore 0.16 is the required probability here.

b) For a normal distribution, half of the values are more than the mean. Therefore 0.5 is the required probability here.

c) -8.3 = 5.9 - 2*7.1 that is 2 standard deviations less than the mean.
20.1 = 5.9 + 2*7.1 that is 2 standard deviations more than the mean.

Therefore 95% of the observations lies within 2 standard deviations of the mean.

Therefore 0.95 is the required probability here.

d) 27.2 = 5.9 + 3*7.1 that is 3 standard deviations more than the mean. We know that 99.7% of the observations lies within 3 standard deviations of the mean. therefore (1 - 0.997)/2 = 0.15% of the observations lies outside 3 standard deviations on either side.

Therefore 0.0015 is the required probability here.