The rates of return on the investments in a large portfolio had mound shapeddistribution, with a mean of 20% and a standard deviation of 10%.

a. What proportion of the investments had a return of between 10% and 30%? b. What proportion of investments had a return that was either less than 10% or morethan 30%?

a) 68% of the investments had a return of between 10% and 30%.

b) 32% of investments had a return that was either less than 10% or morethan 30%.

Step-by-step explanation:

We can use the Empirical Rule to solve this question:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean. This also means that 32% of the measures are more than 1 standard deviation from the mean.

95% of the measures are within 2 standard deviation of the mean. This also means that 5% of the measures are more than 2 standard deviations from the mean.

99.7% of the measures are within 3 standard deviations of the mean. This also means that 0.3% of the measures are more than 3 standard deviations from the mean.

In this problem, we have that:

Mean = 20%.

Standard deviation = 10%.

a. What proportion of the investments had a return of between 10% and 30%?

10 is the mean subtracted by 1 one standard deviation

30 is one standard deviation added to the mean.

So 10 and 30 are within 1 standard deviation of the mean. So 68% of the investments had a return of between 10% and 30%.

b. What proportion of investments had a return that was either less than 10% or morethan 30%?

This is the proportion of investments that were farther than one standard deviation of the mean.

By the Empirical Rule, 32% of investments had a return that was either less than 10% or morethan 30%.