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Questions and Answers of Statistics

In the capital budgeting model in Figure 14.40, investment 4 has the largest ratio of NPV to cash requirement, but it is not selected in the optimal solution. How much NPV is lost if Tatham is
As it currently stands, investment 7 in the capital budgeting model in Figure 14.40 has the lowest ratio of NPV to cash requirement, 2.5. Keeping this same ratio, can you change the cash requirement
Expand the capital budgeting model in Figure 14.40 so that there are now 20 possible investments. You can make up the data on cash requirements, NPVs, and the budget. However, use the following
Suppose in the capital budgeting model in Figure 14.40 that each investment requires $2000 during year 2 and only $5000 is available for investment during year 2.a. Assuming that available money
How difficult is it to expand the Great Threads model to accommodate another type of clothing? Answer by assuming that the company can also produce sweatshirts. The rental cost for sweatshirt
Referring to the previous problem, if it is optimal for the company to produce sweatshirts, use SolverTable to see how much larger the fixed cost of sweatshirt machinery would have to be before the
In the Great Threads model, the production quantities in row 16 were not constrained to be integers. Presumably, any fractional values could be safely rounded to integers. See whether this is true.
In the optimal solution to the Great Threads model, the labor hour and cloth constraints are both binding—the company is using all it has. a. Use SolverTable to see what happens to the optimal
In the optimal solution to the Great Threads model, no pants are produced. Suppose Great Threads has an order for 300 pairs of pants that must be produced. Modify the model appropriately and use
In the original Western Airlines set-covering model in Figure 14.52, we assumed that each city must be covered by at least one hub. Suppose that for added flexibility in flight routing, Western
In the original Western Airlines set-covering model in Figure 14.52, we used the number of hubs as the objective to minimize. Suppose instead that there is a fixed cost of locating a hub in any city,
Set-covering models such as the original Western Airlines model in Figure 14.52 often have multiple optimal solutions. See how many alternative optimal solutions you can find. Of course, each must
How hard is it to expand a set-covering model to accommodate new cities? Answer this by modifying the model in Figure 14.55. Add several cities that must be served: Memphis, Dallas, Tucson,
The models in this section are often called combinatorial models because each solution is a combination of the various 0s and 1s, and there are only a finite number of such combinations. For the
Make up an example, as described in Problem 54, with 20 possible investments. However, do it so that the ratios of NPV to cash requirement are in a very tight range, from 3.0 to 3.2. Then use Solver
In the Great Threads model, we found an upper bound on production of any clothing type by calculating the amount that could be produced if all of the resources were devoted to this clothing type.a.
In the last sheet of the finished version of the Fixed Cost Manufacturing file, we illustrated one way to model the Great Threads problem with IF functions, but saw that this approach doesn’t work.
In the peak-load pricing model, the demand functions have positive and negative coefficients of prices. The negative coefficients indicate that as the price of a product increases, demand for that
In the peak-load pricing model, we assumed that the capacity level is a decision variable. Assume now that capacity has already been set at 30 kwh. Change the model appropriately and run Solver. Then
For each of the following, answer whether it makes sense to multiply the matrices of the given sizes. In each case where it makes sense, demonstrate an example in Excel, where you make up the
Add a new stock, stock 4, to the portfolio optimization model. Assume that the estimated mean and standard deviation of return for stock 4 are 0.125 and 0.175, respectively. Also, assume the
In the portfolio optimization model, stock 2 is not in the optimal portfolio. Use SolverTable to see whether it ever enters the optimal portfolio as its correlations with stocks 1 and 3 vary.
The stocks in the portfolio optimization model are all positively correlated. What happens when they are negatively correlated? Answer for each of the following scenarios. In each case, two of the
In many cases you can assume that the portfolio return is at least approximately normally distributed. Then you can use Excel’s NORMDIST function as in Chapter 5 to calculate the probability that
A bus company believes that it will need the following numbers of bus drivers during each of the next five years: 60 drivers in year 1; 70 drivers in year 2; 50 drivers in year 3; 65 drivers in year
A pharmaceutical company produces the drug NasaMist from four chemicals. Today, the company must produce 1000 pounds of the drug. The three active ingredients in NasaMist are A, B, and C. By weight,
A bank is attempting to determine where to invest its assets during the current year. At present, $500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual
A fertilizer company blends silicon and nitrogen to produce two types of fertilizers. Fertilizer 1 must be at least 40% nitrogen and sells for $70 per pound. Fertilizer 2 must be at least 70% silicon
LP models are used by many Wall Street firms to select a desirable bond portfolio. The following is a simplified version of such a model. A company is considering investing in four bonds; $1 million
At the beginning of year 1, you have $10,000. Investments A and B are available; their cash flows are shown in the file S14_84.xlsx. Assume that any money not invested in A or B earns interest at an
An oil company produces two types of gasoline, G1 and G2, from two types of crude oil, C1 and C2. G1 is allowed to contain up to 4% impurities, and G2 is allowed to contain up to 3% impurities. G1
The government is auctioning off oil leases at two sites: 1 and 2. At each site 10,000 acres of land are to be auctioned. Cliff Ewing, Blake Barnes, and Alexis Pickens are bidding for the oil.
An automobile company produces cars in Los Angeles and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The costs of shipping a car between various
An oil company produces oil from two wells. Well 1 can produce up to 150,000 barrels per day, and well 2 can produce up to 200,000 barrels per day. It is possible to ship oil directly from the wells
Based on Bean et al. (1987). Boris Milkem’s firm owns six assets. The expected selling price (in millions of dollars) for each asset is given in the file S14_89.xlsx. For example, if asset 1 is
Based on Sonderman and Abrahamson (1985). In treating a brain tumor with radiation, physicians want the maximum amount of radiation possible to bombard the tissue containing the tumors. The
A leading hardware company produces three types of computers: Pear computers, Apricot computers, and Orange computers. The relevant data are given in the file S14_91.xlsx. The equipment cost is a
A food company produces tomato sauce at five different plants. The tomato sauce is then shipped to one of three warehouses, where it is stored until it is shipped to one of the company’s four
You are given the following means, standard deviations, and correlations for the annual return on three potential investments. The means are 0.12, 0.15, and 0.20. The standard deviations are 0.20,
You have $50,000 to invest in three stocks. Let Ri be the random variable representing the annual return on $1 invested in stock i. For example, if Ri = 0.12, then $1 invested in stock i at the
The risk index of an investment can be obtained by taking the absolute values of percentage changes in the value of the investment for each year and averaging them. Suppose you are trying to
Broker Sonya Wong is currently trying to maximize her profit in the bond market. Four bonds are available for purchase and sale at the bid and ask prices shown in the file S14_96.xlsx. Sonya can buy
A financial company is considering investing in three projects. If it fully invests in a project, the realized cash flows (in millions of dollars) will be as listed in the file S14_97.xlsx. For
You are a CFA (chartered financial analyst). An overextended client has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards listed in the
A food company produces two types of turkey cutlets for sale to fast-food restaurants. Each type of cutlet consists of white meat and dark meat. Cutlet 1 sells for $4 per pound and must consist of at
Each hour from 10 A.M. to 7 P.M., a bank receives checks and must process them. Its goal is to process all checks the same day they are received. The bank has 13 check processing machines, each of
An oil company has oil fields in San Diego and Los Angeles. The San Diego field can produce up to 500,000 barrels per day, and the Los Angeles field can produce up to 400,000 barrels per day. Oil is
An electrical components company produces capacitors at three locations: Los Angeles, Chicago, and New York. Capacitors are shipped from these locations to public utilities in five regions of the
Based on Bean et al. (1988). The owner of a shopping mall has 10,000 square feet of space to rent and wants to determine the types of stores that should occupy the mall. The minimum number and
It is currently the beginning of 2010. A city (labeled C for convenience) is trying to sell municipal bonds to support improvements in recreational facilities and highways. The face values (in
This problem deals with strategic planning issues for a large company.18 The main issue is planning the company’s production capacity for the coming year. At issue is the overall level of capacity
Kate Torelli, a security analyst for Lion Fund, has identified a gold-mining stock (ticker symbol GMS) as a particularly attractive investment. Torelli believes that the company has invested wisely
Use the RAND function and the Copy command to generate a set of 100 random numbers.a. What fraction of the random numbers are smaller than 0.5?b. What fraction of the time is a random number less
Use Excel’s functions (not @RISK) to generate 1000 random numbers from a normal distribution with mean 100 and standard deviation 10. Then freeze these random numbers.a. Calculate the mean and
Use @RISK to draw a uniform distribution from 400 to 750. Then answer the following questions.a. What are the mean and standard deviation of this distribution?b. What are the 5th and 95th percentiles
Use @RISK to draw a normal distribution with mean 500 and standard deviation 100. Then answer the following questions.a. What is the probability that a random number from this distribution is less
Use @RISK to draw a triangular distribution with parameters 300, 500, and 900. Then answer the following questions.a. What are the mean and standard deviation of this distribution?b. What are the 5th
Use @RISK to draw a binomial distribution that results from 50 trials with probability of success 0.3 on each trial, and use it to answer the following questions.a. What are the mean and standard
Use @RISK to draw a triangular distribution with parameters 200, 300, and 600. Then superimpose a normal distribution on this drawing, choosing the mean and standard deviation to match those from the
We all hate to keep track of small change. By using random numbers, it is possible to eliminate the need for change and give the store and the customer a fair deal. This problem indicates how it
A company is about to develop and then market a new product. It wants to build a simulation model for the entire process, and one key uncertain input is the development cost. For each of the
Continuing the preceding problem, suppose that another key uncertain input is the development time, which is measured in an integer number of months. For each of the following scenarios, choose an
Suppose you own an expensive car and purchase auto insurance. This insurance has a $1000 deductible, so that if you have an accident and the damage is less than $1000, you pay for it out of your
In August of the current year, a car dealer is trying to determine how many cars of the next model year to order. Each car ordered in August costs $20,000. The demand for the dealer’s next year
In the Walton Bookstore example, suppose that Walton receives no money for the first 50 excess calendars returned but receives $2.50 for every calendar after the first 50 returned. Does this change
A sweatshirt supplier is trying to decide how many sweatshirts to print for the upcoming NCAA basketball championships. The final four teams have emerged from the quarterfinal round, and there is now
In the Walton Bookstore example with a discrete demand distribution, explain why an order quantity other than one of the possible demands cannot maximize the expected profit.
If you add several normally distributed random numbers, the result is normally distributed, where the mean of the sum is the sum of the individual means, and the variance of the sum is the sum of the
In Problem 11 from the previous section, we stated that the damage amount is normally distributed. Suppose instead that the damage amount is triangularly distributed with parameters 500, 1500, and
Continuing the previous problem, assume, as in Problem 11, that the damage amount is normally distributed with mean $3000 and standard deviation $750. Run @RISK with 5000 iterations to simulate the
In Problem 12 of the previous section, suppose that the demand for cars is normally distributed with mean 100 and standard deviation 15. Use @RISK to determine the “best” order quantity—in this
Use @RISK to analyze the sweatshirt situation in Problem 14 of the previous section. Do this for the discrete distributions given in the problem. Then do it for normal distributions. For the normal
Although the normal distribution is a reasonable input distribution in many situations, it does have two potential drawbacks: (1) It allows negative values, even though they may be extremely
The Fizzy Company produces six-packs of soda cans. Each can is supposed to contain at least 12 ounces of soda. If the total weight in a six-pack is less than 72 ounces, Fizzy is fined $100 and
When you use @RISK’s correlation feature to generate correlated random numbers, how can you verify that they are correlated? Try the following. Use the RISKCORRMAT function to generate two normally
Repeat the previous problem, but make the correlation between the two inputs equal to –0.7. Explain how the results change.
Repeat Problem 23, but now make the second input variable triangularly distributed with parameters 50, 100, and 500. This time, verify not only that the correlation between the two inputs is
Suppose you are going to invest equal amounts in three stocks. The annual return from each stock is normally distributed with mean 0.01 (1%) and standard deviation 0.06. The annual return on your
The effect of the shapes of input distributions on the distribution of an output can depend on the output function. For this problem, assume there are 10 input variables. The goal is to compare the
The Business School at State University currently has three parking lots, each containing 155 spaces. Two hundred faculty members have been assigned to each lot. On a peak day, an average of 70% of
Six months before its annual convention, the American Medical Association must determine how many rooms to reserve. At this time, the AMA can reserve rooms at a cost of $150 per room. The AMA
You have made it to the final round of the show Let’s Make a Deal. You know that there is a $1 million prize behind either door 1, door 2, or door 3. It is equally likely that the prize is behind
A new edition of a very popular textbook will be published a year from now. The publisher currently has 2000 copies on hand and is deciding whether to do another printing before the new edition comes
A hardware company sells a lot of low-cost, high volume products. For one such product, it is equally likely that annual unit sales will be low or high. If sales are low (60,000), the company can
W. L. Brown, a direct marketer of women’s clothing, must determine how many telephone operators to schedule during each part of the day. W. L. Brown estimates that the number of phone calls
Assume that all of a company’s job applicants must take a test, and that the scores on this test are normally distributed. The selection ratio is the cutoff point used by the company in its hiring
Lemington’s is trying to determine how many Jean Hudson dresses to order for the spring season. Demand for the dresses is assumed to follow a normal distribution with mean 400 and standard
Dilbert’s Department Store is trying to determine how many Hanson T-shirts to order. Currently the shirts are sold for $21, but at later dates the shirts will be offered at a 10% discount, then a
The annual return on each of four stocks for each of the next five years is assumed to follow a normal distribution, with the mean and standard deviation for each stock, as well as the correlations
It is surprising (but true) that if 23 people are in the same room, there is about a 50% chance that at least two people will have the same birthday. Suppose you want to estimate the probability that
Simulation can be used to illustrate a number of results from statistics that are difficult to understand with non-simulation arguments. One is the famous central limit theorem, which says that if
In statistics we often use observed data to test a hypothesis about a population or populations. The basic method uses the observed data to calculate a test statistic (a single number), as discussed
A technical note in the discussion of @RISK indicated that Latin Hypercube sampling is more efficient than Monte Carlo sampling. This problem allows you to see what this means. The file S15_44.xlsx
We are continually hearing reports on the nightly news about natural disasters—droughts in Texas, hurricanes in Florida, floods in California, and so on. We often hear that one of these was the
In Example 16.1, the possible profits vary from negative to positive for each of the 10 possible bids examined.a. For each of these, use @RISK’s RISKTARGET function to find the probability that
If the number of competitors in Example 16.1 doubles, how does the optimal bid change?
Referring to Example 16.1, if the average bid for each competitor stays the same, but their bids exhibit less variability, does Miller’s optimal bid increase or decrease? To study this question,
In Example 16.2, the gamma distribution was used to model the skewness to the right of the lifetime distribution. Experiment to see whether the triangular distribution could have been used instead.
See how sensitive the results in Example 16.2 are to the following changes. For each part, make the change indicated, run the simulation, and comment on any differences between your outputs and the
Rerun the new car simulation from Example 16.4, but now introduce uncertainty into the fixed development cost. Let it be triangularly distributed with parameters $600 million, $650 million, and $850

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