Question

In the last quarter of​ 2007, a group of 64 mutual funds had a mean return of 4.8​% with a standard deviation of 5.8​%. 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 10.6​% or more ​b) Returns of 4.8​% or less ​c) Returns between negative 12.6​% and 22.2​% ​d) Returns of more than 16.4​%

Answer 1

Using the 68-95-99.7 rule for the mutual fund returns, approximately 16% of funds are expected to have returns of 10.6% or more, 50% at 4.8% or less, 99.7% between -12.6% and 22.2%, and about 2.5% with returns of more than 16.4%.

Step-by-step explanation:

To answer the student's question regarding the expected percentage of mutual funds in each region based on their returns, we should first draw a normal distribution curve and apply the 68-95-99.7 rule. This rule tells us that approximately 68% of the values lie within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Given the mean return of 4.8% and a standard deviation of 5.8%, we can calculate the following:

a) Returns of 10.6% or more would be more than one standard deviation above the mean. The rule states that about 16% (100%-84%) of the values will lie above one standard deviation, so that's the percentage of funds expected to have returns of 10.6% or more.

b) Returns of 4.8% or less would include approximately 50% of the funds, as the mean divides the normal distribution in half.

c) Returns between -12.6% and 22.2% would span three standard deviations on both sides of the mean, which includes about 99.7% of the funds.

d) Returns of more than 16.4% would be more than two standard deviations above the mean. Given that 95% of the values are within two standard deviations, this means that approximately 2.5% (100%-97.5%) of the funds would have returns of more than 16.4%.

By applying the 68-95-99.7 rule, we can estimate the percentages without using technology for a more precise calculation.

Answer 2

a) 16%; b) 50%; c) 99.7%; d) 2.5%

Step-by-step explanation:

In a normal curve, the empirical rule states that 68% of data falls within 1 standard deviation of the mean. This means for this problem, 68/2 = 34% of data falls from

4.8-5.8 = -1 to 4.8, and 34% falls from 4.8 to

4.8+5.8 = 10.6.

95% of data falls within 2 standard deviations of the mean. This includes the 68%; this means this leaves 95-68 = 27/2 = 13.5% to fall from

-1-5.8 = -6.8 to -1, and 13.5% falls from 10.6 to

10.6+5.8 = 16.4.

99.7% of data falls within 3 standard deviations of the mean. This includes the 95%; this means this leaves 99.7-95 = 4.7/2 = 2.35% to fall from

-6.8-5.8 = -12.6 to -6.8, and 2.35% falls from 16.4 to

16.4+5.8 = 22.2.

This leaves 100-99.7 = 0.3/2 = 0.15% to fall from the left end to -12.6, and 0.15% to fall from 22.2 to the right end.

For part a,

For returns of 10.6 or more, we would add everything above this value:

13.5+2.35+0.15 = 16%.

For part b,

Since 4.8 is the mean, 50% of data falls below this.

For part c,

-12.6 is 3 standard deviations from the mean, and 22.2 is 3 standard deviations from the mean. This means that 99.7% of the data falls between these values.

For part d,

We add together all values above 16.4: 2.35+0.15 = 2.5%