Midterm Examination Answer all 5 questions and show all your work. All work must be shown. ________________________________________________________________________ 1. The average age for a person getting married for the first time is 26 years (U.S. News and World Report, June 6, 1994). Assume the ages for first marriages have a normal distribution with a standard deviation of four years. a) What is the probability that a person getting married for the first time is younger than 23 years of age? b) What is the probability that a person getting married for the first time is in his or her twenties? c) 90% of people getting married for the first time get married before what age? 2. A management company must first decide whether to undertake a market research survey. If the market research study is conducted, the outcome will either be favorable (F) or unfavorable (U). Assume there are only two decision alternatives, D1 and D2, and two states of nature, S1 and S2. The payoff table showing profit is as follows: State of Nature Decision Alternative S1 S2 D1 100 300 D2 400 200 a) Using the following probabilities, what is the optimal decision strategy? P(F) = 0.56 P(S1/F) = 0.57 P(S1/U) = 0.18 P(S1) = 0.40 P(U) = 0.44 P(S2/F) = 0.43 P(S2/U) = 0.82 P(S2) = 0.60 b) Find the Efficiency of this market research. What would be your recommendation? c) How much will this company be willing to pay for this market research? 3. The demand for Carolina Industries” product varies greatly from month to month. Based on the past two years of data, the following probability distribution shows the company’s monthly demand: Unit Demand Probability 300 0.20 400 0.30 500 0.35 600 0.15 a. If the company places monthly orders equal to the expected value of the monthly demand, what should Carolina’s monthly order quantity be for this product? b. Assume that each unit demanded generates $70 in revenue and that each unit ordered costs $50. How much will the company gain or lose in a month if it places an order based on your answer in part (a) and the actual demand for the item is 300 units? c. What are the variance and standard deviation for the number of units demanded? 4. Lawson’s Department Store faces a buying decision for a seasonal product for which demand can be high or low. The purchaser for Lawson’s can order 1, 2, or 3 lots of the product before the season begins but cannot reorder later. Profit projections (in thousands of dollars) are shown. State of Nature Decision Alternative High Demand (S1) Low Demand (S2) Order 1 lot, D1 60 50 Order 2 lots, D2 80 30 Order 3 lots, D3 100 10 P(Sj) 0.4 0.6 a) Find the optimal decision using the expected value approach. b) Use the expected loss opportunity method to find the Expected Value of Perfect Information. c) Conduct a graphical sensitivity analysis identifying the decisions for the different P(S1). d) For the optimal solution, given the current probabilities of the state of nature, what would be the range of optimality (that is the range of values for its payoff where that decision alternative will still remain optimal) for its payoffs compared to the closest Expected Value. 5. A survey on British Social Attitudes asked respondents if they had ever boycotted goods for ethical reasons (Statesman, January 28, 2008). The survey found that 23% of the respondents have boycotted goods for ethical reasons. a) In a sample of six British citizens, what is the probability that two have ever boycotted goods for ethical reasons? b) In a sample of six British citizens, what is the probability that at least two respondents have boycotted goods for ethical reasons? c) In a sample of ten British citizens, what is the probability that between 3 and 6 have boycotted goods for ethical reasons? d) In a sample of ten British citizens, what is the expected number of people that have boycotted goods for ethical reasons? Also find the standard deviation.
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