Essentials of Modern Business Statistics with Microsoft® Excel®
Exercises
25. Given below is a bivariate distribution for the random variables x and y
f (x, y) x y
.2 50 80
.5 30 50
.3 40 60
a. Compute the expected value and the variance for x and y.
b. Develop a probability distribution for x 1 y.
c. Using the result of part (b), compute E(x 1 y) and Var(x 1 y).
d. Compute the covariance and correlation for x and y. Are x and y positively related,
negatively related, or unrelated?
e. Is the variance of the sum of x and y bigger than, smaller than, or the same as the
sum of the individual variances? Why?
26. A person is interested in constructing a portfolio. Two stocks are being considered. Let x 5 percent return for an investment in stock 1, and y 5 percent return for an investment in stock 2. The expected return and variance for stock 1 are E(x) 5 8.45% and Var(x) 5 25. The expected return and variance for stock 2 are E(y) 5 3.20% and Var(y) 5 1. The covariance between the returns is sxy 5 23.
a. What is the standard deviation for an investment in stock 1 and for an investment in stock 2? Using the standard deviation as a measure of risk, which of these stocks is
the riskier investment?
b. What is the expected return and standard deviation, in dollars, for a person who
invests $500 in stock 1?
c. What is the expected percent return and standard deviation for a person who con-
structs a portfolio by investing 50% in each stock?
d. What is the expected percent return and standard deviation for a person who con-
structs a portfolio by investing 70% in stock 1 and 30% in stock 2?
e. Compute the correlation coefficient for x and y and comment on the relationship
between the returns for the two stocks.
27. Canadian Restaurant Ratings. The Chamber of Commerce in a Canadian city has conducted an evaluation of 300 restaurants in its metropolitan area. Each restaurant received a rating on a 3-point scale on typical meal price (1 least expensive to 3 most expensive)and quality (1 lowest quality to 3 greatest quality). A crosstabulation of the rating data is shown. Forty-two of the restaurants received a rating of 1 on quality and 1 on meal price, 39 of the restaurants received a rating of 1 on quality and 2 on meal price, and so on. Forty-eight of the restaurants received the highest rating of 3 on both quality and meal price
Quality (x) 1 2 3 Total
1 42 39 3 84
2 33 63 54 150
3 3 15 48 66
Total 78 117 105 300
a. Develop a bivariate probability distribution for quality and meal price of a ran-
domly selected restaurant in this Canadian city. Let x 5 quality rating and y 5 meal
price.
b. Compute the expected value and variance for quality rating, x.
c. Compute the expected value and variance for meal price, y.
d. The Var(x + y) 5 1.6691. Compute the covariance of x and y. What can you
say about the relationship between quality and meal price? Is this what you
would expect?
e. Compute the correlation coefficient between quality and meal price? What is the
strength of the relationship? Do you suppose it is likely to find a low-cost restaurant
in this city that is also high quality? Why or why not?
28. Printer Manufacturing Costs. PortaCom has developed a design for a high-quality
portable printer. The two key components of manufacturing cost are direct labor and
parts. During a testing period, the company has developed prototypes and conducted
extensive product tests with the new printer. PortaCom’s engineers have developed the following bivariate probability distribution for the manufacturing costs. Parts cost (in dollars) per printer is represented by the random variable x and direct labor cost (in dollars) per printer is represented by the random variable y. Management would like to use this probability distribution to estimate manufacturing costs.
Direct Labor (y)
Parts (x) 43 45 48 Total
85 0.05 0.2 0.2 0.45
95 0.25 0.2 0.1 0.55
Total 0.30 0.4 0.3 1.00
a. Show the marginal distribution of direct labor cost and compute its expected value,
variance, and standard deviation.
b. Show the marginal distribution of parts cost and compute its expected value, vari-
ance, and standard deviation.
c. Total manufacturing cost per unit is the sum of direct labor cost and parts cost.
Show the probability distribution for total manufacturing cost per unit.
d. Compute the expected value, variance, and standard deviation of total manufactur-
ing cost per unit.
e. Are direct labor and parts costs independent? Why or why not? If you conclude that
they are not, what is the relationship between direct labor and parts cost?
f. PortaCom produced 1,500 printers for its product introduction. The total manufacturing cost was $198,350. Is that about what you would expect? If it is higher or lower, what do you think may be the reason?
29. Investment Portfolio of Index Fund and Core Bonds Fund. J.P. Morgan Asset Management publishes information about financial investments. Between 2002 and 2011 the expected return for the S&P 500 was 5.04% with a standard deviation of 19.45% and the expected return over that same period for a core bonds fund was 5.78% with a standard deviation of 2.13% (J.P. Morgan Asset Management, Guide to the Markets).
The publication also reported that the correlation between the S&P 500 and core bonds is 2.32. You are considering portfolio investments that are composed of an S&P 500 index fund and a core bonds fund.
a. Using the information provided, determine the covariance between the S&P 500
and core bonds.
b. Construct a portfolio that is 50% invested in an S&P 500 index fund and 50% in
a core bonds fund. In percentage terms, what are the expected return and standard
deviation for such a portfolio?
c. Construct a portfolio that is 20% invested in an S&P 500 index fund and 80%
invested in a core bonds fund. In percentage terms, what are the expected return and
standard deviation for such a portfolio?
d. Construct a portfolio that is 80% invested in an S&P 500 index fund and 20%
invested in a core bonds fund. In percentage terms, what are the expected return and
standard deviation for such a portfolio?
e. Which of the portfolios in parts (b), (c), and (d) has the largest expected return?
Which has the smallest standard deviation? Which of these portfolios is the best
investment alternative?
f. Discuss the advantages and disadvantages of investing in the three portfolios in
parts (b), (c), and (d). Would you prefer investing all your money in the S&P 500
index, the core bonds fund, or one of the three portfolios? Why?
30. Investment Fund Including REITs. In addition to the information in exercise29 on the S&P 500 and core bonds, J.P. Morgan Asset Management reported that the
expected return for real estate investment trusts (REITs) was 13.07% with a standard deviation of 23.17% (J.P. Morgan Asset Management, Guide to the Markets). The cor-
relation between the S&P 500 and REITs is .74 and the correlation between core bonds and REITs is 2.04. You are considering portfolio investments that are composed of an S&P 500 index fund and REITs as well as portfolio investments composed of a core bonds fund and REITs.
a. Using the information provided here and in exercise 29, determine the covariance
between the S&P 500 and REITs and between core bonds and REITs.
b. Construct a portfolio that is 50% invested in an S&P 500 fund and 50% invested in
REITs. In percentage terms, what are the expected return and standard deviation for
such a portfolio?
c. Construct a portfolio that is 50% invested in a core bonds fund and 50% invested in
REITs. In percentage terms, what are the expected return and standard deviation for
such a portfolio?
d. Construct a portfolio that is 80% invested in a core bonds fund and 20% invested in
REITs. In percentage terms, what are the expected return and standard deviation for
such a portfolio?
e. Which of the portfolios in parts (b), (c), and (d) would you recommend to an
aggressive investor? Which would you recommend to a conservative investor?
Why?
