Problem Set #2

Probability

1.  According to Investment Digest ("Diversification and the Risk/Reward Relationship", Spring 2011, 1-3), the mean of the annual return for common stocks from 1936 to 2011 was 15.4%, and the standard deviation of the annual return was 30.5%.  During the same 74-year time span, the mean of the annual return for long-term government bonds was 5.2%, 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.

1. Find the probability that the return for common stocks will be greater than 12%.
2. 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

Confidence Interval Estimation

2.  Compute a 99% confidence interval for the population mean, based on the sample 52, 50, 51, 54, 55, 52, and 53.  Change the number from 53 to 14 and recalculate the confidence interval.  Using the results, describe the effect of an outlier or extreme value on the confidence interval.

Hypothesis Testing

3.  The director of Heating and Oil at Mitchell Brothers Company is concerned about the high cost of home heating oil being offered to their customers for the upcoming Fall season in Boston, MA.  The company has no possibility of modifying the oil price under the current economic conditions.  He believes that the company should offer a low cost maintenance contract to foster good will.  The director decides to survey a sample of the surrounding towns to see what people are paying for similar home maintenance contracts.  The telephone sample of 20 homes indicated that the customers are paying X (bar) = \$285.4 and s = \$42.20 for a yearly contract.

a. Using the 0.05 level of significance, is there evidence that the population mean is above \$290?

b. What is your answer in (a) if s = \$60 and the 0.10 level of significance is used?

c. What is your answer in (a) if X (bar) = \$299.00 and s = \$20.20?

4.  A large hat manufacturer, MICHAELLA HATS, is concerned that the mean weight of their classic Riding hat is not greater than 3.0 pounds.  It can be assumed that the population standard deviation is 1.2 pounds based on past experience.  A sample of 350 hats is selected and the sample mean is 3.15 pounds.  Using a level of significance of .10, is there evidence that the population mean weight of the hats is greater than 3.0 pounds? Fully explain your answer.