90-760: Decision & Risk Modeling Midterm, Spring 2017Do not turn the page until you are told to do so. Write your answers below each question.Note how much each problem is worth and budget your time. Total points possible: 115Name: ______________________________Q1: _________ out of 6 possible pointsQ2: _________ out of 12 possible pointsQ3: _________ out of 16 possible pointsQ4: _________ out of 20 possible pointsQ5: _________ out of 18 possible pointsQ6: _________ out of 9 possible pointsQ7: _________ out of 24 possible pointsQ8: _________ out of 10 possible pointsTotal Possible Points: 115Question #1: (6 points)With regard to some random quantity of interest Y, we contrasted the 90% confidence interval for its mean E[Y]and the 90% confidence interval for Y itself. Assuming you have built a Monte Carlo simulation of thisimportant outcome Y, concisely and precisely explaina) How you would create each 90% CI from the simulation resultsb) The conditions under which one vs. the other would be more relevant for decisions makers thinkingabout Y.Question #2: (12 points)Suppose you had built a spreadsheet that models some outcome of interest Y as a function f(X) of one or moreunknown inputs X. We noted in class that E[f(X)] is not necessarily the same as f(E[X]). a) How would you compute f(E[X]) with your spreadsheet?b) How would you compute E[f(X)] with your spreadsheet?c) Under what conditions would they be same?d) When they differ, which is more informative about the central tendency of the unknown quantity ofinterest Y?Question #3: (16 points)I’ve posted below a simplified version of a spreadsheet we used in class for doing time series forecasting,specifically using regression with a quadratic trend supplemented with additive seasonal adjustments. It appliesthe same method you would have used on the sea ice data in the first homework.a) What formula belongs in Cells F5? __________________________________________b) What formula belongs in cell G5? __________________________________________c) What formula belongs in cell H25 if you want to be able to copy that formula down to all of the otherrows? __________________________________________d) How is the -5409 in cell K5 calculated? (You can simply give the Excel formula, but since it is trickierthan most, you can get full credit for a clear English sentence explaining what that formula does.)Question #4: (20 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 rows not shown (rows 11-24) to save space.a) How many variables were used to make these forecasts?b) How many observations were there in the training and classification data sets, respectively?c) Fill in the 9 missing cells marked in grey, including the % correct for the training sample. (If theshading is hard to read on the printed exam, these are cells E26, E27, E32, B33, B34, D34, B36, E42, &E43.)d) What is the meaning of the number 85.00 in cell B5?e) What is the meaning of the number 2.97 in cell B9?Question #5: (18 points)Below is a 2 X 2 “confusion matrix” for an (imperfect) cancer screening test. Fill in the nine quantities belowthe matrix. Note: As I warned in recitation, I have jumbled up rows and columns relative to the display I usedin class, so you need to think through what is a “true positive” vs. a “false positive” from first principles, notjust by remembering that this or that quantity is associated with the lower right or lower left cell. To be clear,the actual condition we are testing for is cancer being present. Question #6: (9 points)I modified the symphony net revenue model we used in class to make the random variables follow a triangledistribution and adjusted a number of the parameters. The result of 10,000 simulation trials is show below.a) What Excel formula belongs in Cell I2? __________________________________________?b) What Excel formula belongs in Cell I17? __________________________________________?c) Based on this simulation, approximately what is the probability that the symphony will have negativenet revenues?Question #7: (24 points)Below is a screen shot of a payoff matrix indicating the cost in dollars for each of 8 alternatives under 5 possiblestates of nature, along with its associated calculations & risk-return frontier plotting standard deviation vs.expected value (i.e., expected cost). a) What numbers belong in Cells I4, K4? __________________________________________b) What Excel formula belongs in Cell I5? __________________________________________c) What Excel formula belongs in cell J5? __________________________________________d) What Excel formula belongs in cell L5? __________________________________________e) What option should a risk neutral decision maker prefer?f) Draw a smiley face in the most desirable corner of the risk-return frontier.g) Circle the dots corresponding to options that are on the efficient frontier.h) Suppose a ninth alternative emerged that had an expected value of 50 and standard deviation of 20.Mark with a star where that alternative would appear on the graph.Question #8: (10 points)Neatly sketch the cumulative risk profiles for Options A (Solid line) and B (Dashed line) if big numbers aregood and: - Option A pays $5 if a (biased) coin with probability 0.6 of showing heads comes up heads and $15 if itcomes up tails. - Option B pays off $10 for sure.Also, state any dominance (statewise, 1st order stochastic, or 2nd order stochastic) that Option A has over OptionB or vice versa and what that implies for decision makers with relevant risk attitudes.