Book cover for Statistics for Business and Economics

Statistics for Business and Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

ISBN #9780324365054

10th Edition

999 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

In decision analysis, tools such as payoff tables and decision trees are essential for structuring and solving complex decision problems under uncertainty. By calculating expected values for each decision alternative and incorporating probability assessments, decision makers can identify the optimal strategy. Furthermore, evaluating the expected value of additional information (both perfect and sample-based) helps determine whether further investment in research is worthwhile. Overall, this approach ensures a systematic, quantitative basis for making informed decisions under risk.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature. a. $\quad$ Construct a decision tree for this problem. b. Suppose that the decision maker obtains the probabilities $P\left(s_{1}\right)=.65, P\left(s_{2}\right)=.15$ and $P\left(s_{3}\right)=.20 .$ Use the expected value approach to determine the optimal decision.

Example 2

A decision maker faced with four decision alternatives and four states of nature develops the following profit payoff table. The decision maker obtains information that enables the following probabilities assessments: $P\left(s_{1}\right)=.5, P\left(s_{2}\right)=.2, P\left(s_{3}\right)=.2,$ and $P\left(s_{1}\right)=.1$ a. Use the expected value approach to determine the optimal solution. b. Now assume that the entries in the payoff table are costs. Use the expected value approach to determine the optimal decision.

Example 3

Hudson Corporation is considering three options for managing its data processing operation: continue with its own staff, hire an outside vendor to do the managing (referred to as outsourcing), or use a combination of its own staff and an outside vendor. The cost of the operation depends on future demand. The annual cost of each option (in thousands of dollars) depends on demand as follows. a. If the demand probabilities are $.2, .5,$ and $.3,$ which decision alternative will minimize the expected cost of the data processing operation? What is the expected annual cost associated with your recommendation? b. What is the expected value of perfect information?

Example 4

Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full price service using the company's new fleet of jet aircraft and a discount service using smaller capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based upon two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars). a. What is the decision to be made, what is the chance event, and what is the consequence for this problem? How many decision alternatives are there? How many outcomes are there for the chance event? b. Suppose that management of Myrtle Air Express believes that the probability of strong demand is .7 and the probability of weak demand is $.3 .$ Use the expected value approach to determine an optimal decision. c. Suppose that the probability of strong demand is .8 and the probability of weak demand is .2. What is the optimal decision using the expected value approach?

Example 5

The distance from Potsdam to larger markets and limited air service have hindered the town in attracting new industry. Air Express, a major overnight delivery service, is considering establishing a regional distribution center in Potsdam. But Air Express will not establish the center unless the length of the runway at the local airport is increased. Another candidate for new development is Diagnostic Research, Inc. (DRI), a leading producer of medical testing equipment. DRI is considering building a new manufacturing plant. Increasing the length of the runway is not a requirement for DRI, but the planning commission feels that doing so will help convince DRI to locate their new plant in Potsdam. Assuming that the town lengthens the runway, the Potsdam planning commission believes that the probabilities shown in the following table are applicable. $\begin{array}{lcc} & \text { DRI Plant } & \text { No DRI Plant } \\ \text { Air Express Center } & .30 & .10 \\ \text { No Air Express Center } & .40 & .20\end{array}$ For instance, the probability that Air Express will establish a distribution center and DRI will build a plant is .30. The estimated annual revenue to the town, after deducting the cost of lengthening the runway, is as follows: $$\begin{array}{lcr} & \text { DRI Plant } & \text { No DRI Plant } \\ \text { Air Express Center } & \$ 600,000 & \$ 150,000 \\ \text { No Air Express Center } & \$ 250,000 & -\$ 200,000 \end{array}$$ If the runway expansion project is not conducted, the planning commission assesses the probability DRI will locate their new plant in Potsdam at .6; in this case, the estimated annual revenue to the town will be $\$ 450,000$. If the runway expansion project is not conducted and DRI does not locate in Potsdam, the annual revenue will be $\$ 0$ since no cost will have been incurred and no revenues will be forthcoming. a. What is the decision to be made, what is the chance event, and what is the consequence? b. Compute the expected annual revenue associated with the decision alternative to lengthen the runway. c. Compute the expected annual revenue associated with the decision alternative to not lengthen the runway. d. Should the town elect to lengthen the runway? Explain. e. Suppose that the probabilities associated with lengthening the runway were as follows: $\begin{array}{lcc} & \text { DRI Plant } & \text { No DRI Plant } \\ \text { Air Express Center } & .40 & .10 \\ \text { No Air Express Center } & .30 & .20\end{array}$ What effect, if any, would this change in the probabilities have on the recommended decision?
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