Probability

1. According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 15.4%, and the standard deviation of the annual return was 24.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 6.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.

a. Find the probability that the return for common stocks will be greater than 9%.

b. Find the probability that the return for common stocks will be greater than 25%.

Hint: There are many ways to attack this problem in the HW. If you would like the normal distribution table so you can draw the pictures (my preferred way of learning) then I suggest you bookmark this site:

http://www.statsoft.com/textbook/sttable.html

Solution:

a. Find the probability that the return for common stocks will be greater than 9%.

Mean = 15.4% standard deviation = 24.5%.

[pic]

The probability that the return for common stocks will be greater than 9% is 0.6026.

b. Find the probability that the return for common stocks will be greater than 25%.

Mean = 15.4% standard deviation = 24.5%.

[pic]

The probability that the return for common stocks will be greater than 25% is 0.3483.

Confidence Interval Estimation

2. Compute a 95% confidence interval for the population mean, based on the sample 25, 27, 23, 24, 25, 24, and 59. Change the number from 59 to 24 and recalculate the confidence interval. Using the results, describe the effect of an outlier or extreme value on the confidence interval.

Solution:

The formula for confidence interval for the population mean

[pic]

| ...