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90-760: Decision & Risk Modeling Midterm Solution, PM Section Spring 2016Question #1: (10 points)Concisely explain the key ideas concerning human judgment & forecasting that are captured by these vignettesdiscussed in the first class. Use neat handwriting. (Verbose or illegible answers will not be graded or will bemarked down sharply.)a) Israeli judges are less likely to grant parole in cases reviewed at the end of the day.b) When guessing the weight of an ox, the average of the guesses of 50 passers-by is often more accuratethan the expert judgment of a butcher. (For non-native English speakers: A butcher is a person whoslaughters animals and sells their meat.)Solution:a) In short, humans are human and so vulnerable to having their judgment and decision-makingswayed by all sorts of irrelevant factors, such as how hungry or tired one is (which in turn areaffected by how long it has been since the last lunch or snack break).b) This illustrates what James Surowiecki calls “The Wisdom of Crowds”. As long as the laypeoplehave access to at least partially independent information and do not influence each other (which iswhat happens when crowds behave unwisely, such as in a panic or stampede), they collectively offer abetter basis for forecasting than any single expert does. Question #2: (15 points)For the exponential smoothing time-series forecasting method, concisely answer the following in neathandwriting. (Verbose or illegible answers will not be graded or will be marked down sharply.)a) How does one find the parameter ?b) What sort of value for would you expect if the system generating the time series had been stable for along time?c) What is the relative weight exponential smoothing puts on the 2nd most recent vs. the 4th most recent datapoint? (You must be precise about the algebraic relationships to get credit.)Solution:a) One optimizes (e.g., with Excel’s Solver) to minimize a measure of error (such as the RMSE) betweenthe model’s “backcast” for past periods and the actual observations’ values, treating as a decisionvariable (or “blue cell”) in that optimization. [You might constrain to be non-negative and lessthan 1.0 and worry if it ends up being larger than 0.5]b) Small .c) w4 = (1 – )2 w2. Question #3: (25 points)I ran discriminant analysis with the DA.xla add-in on a data set concerning job applicants and obtained theresults shown below, with some cells deleted.a) Fill in the 8 missing cells in the confusion matrix & also the % correct for the training sample.b) Which group represents the strongest set of candidates?c) Which predictor seems to be the least useful?d) For which classification observations was the group assignment a “close call”?e) How far is Observation #3 in the training sample from the centroid for group #3?Solution:a)b) Group #3 since it has the most experience and highest GPA. c) Interview scores are nearly identical across groups and so do not appear to provide useful predictiveinformation.d) Observations #1 and/or #4. (So answers of #1, #4, or #1 & #4 are all accepted.) For those twoobservations the distance to the second closest centroid is very close to the distance to the closest centroid. e) 2.58Question #4: (20 points)Below is a screen shot of a payoff matrix indicating the cost in dollars for each of 6alternatives under 4 possiblestates of nature, along with its associated calculations & risk-return frontier plotting standard deviation vs.expected value. a) What number belongs in Cell G4? __________________________________________b) What number belongs in Cell I6? __________________________________________c) Draw a smiley face in the most desirable corner of the risk-return frontier.d) Which option(s) is/are not on this frontier? (Identify them by number.) Solution:a) 4.00b) 0.253c) Smiley face goes in the lower left cornerd) #2 and #4Question #5: (30 points)A 300-seat theater sells concert tickets three ways: (1) Before the season as part of a season’s ticket package atthe equivalent of $50 per ticket, (2) As individual advance tickets at the “normal” price of $80, and (3) At the“student rush” price ($20) on the day of the show. The season includes six different concerts, each performedjust once, and to keep things simple, suppose there is effectively infinite demand for same-day tickets, so noseat goes empty. The theater needs to decide what is the maximum number of the season’s tickets it should bewilling to sell and, hence, how many seats to hold back so they can be sold as individual tickets or same daytickets. Below I have pasted the upper left corner of a Monte Carlo simulation to help the theater make itsdecision, where demand for individual tickets is modeled simply as a discrete uniform random variable betweenthe specified minimum and maximum values (which vary by show). What I have not shown is that the columnsextend to the right covering “Sell now” options from 0 up to 210 in steps of 15, and the rows extend downthrough row 10,019 to allow for 10,000 simulation trials. Below is a screen shot of a payoff matrix with 10 alternatives and 5 states of nature and its associated risk-return frontier plotting standard deviation vs. expected value. Assume big numbers are good.a) What problem or kind of problem that we’ve discussed in class does this resemble? b) What is the formula in Cell G1? __________________________________________c) What is the formula in Cell G2? __________________________________________d) What is the formula in Cell G3? __________________________________________e) What is the formula in Cell D9? __________________________________________f) What is the formula in Cell G9? __________________________________________g) What is the formula in Cell G17? __________________________________________h) What is the formula in Cell G19? __________________________________________Solutiona) This is a variant on a single period inventory problem and, in particular, most closely resembledthe WVTU television station problem in which the station had to decide how many television adsto sell now and how many to hold back until the days leading up to the November election. b) What is the formula in Cell G1? __ =AVERAGE(G20:G10019)______c) What is the formula in Cell G2? ____ =STDEV(G$20:G$10019)____________d) What is the formula in Cell

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