The historical returns on a portfolio had an average return of 19 percent and a standard deviation of 25 percent. Assume that returns on this portfolio follow a bell-shaped distribution. a. Approximately what percentage of returns were greater than 69 percent?b. Approximately what percentage of returns were below –56 percent?

a)

And we can find this probability using the complement rule and the z table or excel:

b)

And we can find this probability using the z table or excel:

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the historical returns of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the complement rule and the z table or excel:

Part b

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability using the z table or excel:

80% of the cookies

step-by-step explanation: