Mock Exam 1

ASC TO 301 Midterm Mock Exam Answers 10/16/23 Module 1 - Optimization 1. Which one of the following is always true about the sensitivity report in linear programming? a. Shadow price is the change in objective value resulting from a unit change in one of the resources. b. If slack is 0 for a given constraint, then the constraint is non-binding. c. Shadow price of a non-binding constraint is zero. d. A and C are both true e. A and B are both true D is the correct answer. A is the definition of shadow price. B is false because if slack is 0, that means RHS-LHS = 0, which is a requirement for a binding constraint. C is true because the shadow price of a non-binding constraint is zero, it has no impact on the objective function value or optimal solution . 2. Below is the sensitivity analysis report for a clothing factory, where the objective is to maximize profit . Cell Name Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease $E$1 T-shirts 20 0 40 50 8 $E$2 Sweatpants 80 0 25 20 20 $E$3 Hoodies 50 0 30 10 6 All else being equal, if the clothing factory increases the per-unit profit of sweatpants to _______, then the change in profit will be _________ and the new profit will be _________. a. $30; $2400; $4700 b. $50; $2000; $6300 c. $40; $1200; $5500 d. None of the above are correct. C is the correct answer. A is incorrect because the change in profit is not: objective coefficient x final value, rather the change in the objective coefficient times the final value. B is incorrect because the change is outside the allowable increase, so we cannot use the sensitivity analysis to determine the new profit. C is correct because the change in profit is

(40-25) * (80) = 1200; and the new profit is: 20*40+80*40+50*30 = 5500. 3. [Daphne] Which of the following statements is correct about linear programming? a. For an infeasible linear program, the optimal solution is always 0 b. An optimal solution is a feasible solution with the largest value of the objective function c. An unbounded linear program may or may not have an optimal solution d. There can only be one optimal solution e. The value of the optimal solution indicates the profit-maximizing per unit price, if price were an objective coefficient. C is the right answer. A is wrong because an infeasible LP has no optimal solution. B is wrong because the optimal solution is a feasible solution with the most favorable value of the objective function, which could be the maximum or minimum point depending on the question. D is wrong because when a z-line has the same slope as a constraint, there can be 2 corner-point optimal solutions. E is wrong because the objective value of the optimal solution indicates the unit amount to produce, not the per unit price. 4. Consider the investment decisions that the CFO of Hugh Industries must go through. She can choose to either build both an airplane hangar [x 1 ] and an airplane factory [x 2 ] or not build at all. Assume that X 1 and X 2 are binary variables (meaning either 0 or 1). How can she model this constraint in a linear programming model? a. X 1 + X 2 = 0 b. X 1 - X 2 = 0 c. X 1 + X 2 = 2 d. X 1 - X 2 = 1 e. X 1 = X 2 (b) is the correct answer, (a) cannot work because if they are both true then the answer is 2, (c) cannot work, because if they are both false then the answer is 0, (d) cannot work because it could never equal 1, and (e) is a correct understanding that would not work in excel. Side note: Putting all variables on the left side for (e) will lead you to (b) the correct answer 5. Which of the following statements is TRUE ? a. When solving a two dimensional LP with a graphical approach, to find the optimal solution we must always move the z-line towards the feasible points that are the farthest away from the origin of the graph. b. A linear program always has at least one corner point feasible solution. c. It is possible to have a single optimal solution when solving an unbounded feasible region of an LP. d. A and C are true. e. B and C are true.

d. When there is no optimal solution, we know the feasible region is unbounded Module 2 - Probability & Decision Analysis 9. Consider a box filled with cartons of milk. In this box, there are sixteen cartons, including 4 strawberries, 6 regular, and 6 chocolate. Suppose I am grabbing two cartons of milk without replacement. With the following events, which properties hold true? X: both cartons of milk is chocolate Y: one carton of milk is strawberry a. Y and X are collectively exhaustive b. X and Y are mutually exclusive c. X and Y are independent d. Two of the above e. All of the above B is the correct answer. These two events are not collectively exhaustive because you can run the experiment without satisfying either X or Y, like getting two regular cartons. They are mutually exclusive because you cannot get both X and Y at the same time. They are not independent because there is no replacement, so the first event impacts the second. Use the following information to answer questions 10 and 11: Depending on certain activities, people enjoy listening to different kinds of music and artists. A recent study on undergraduate students found that if a person was at a “pregame” (a party before a night on the town) 23% of the time they were listening to an artist named Ke$ha. Furthermore, if a person was at the gym or working out, they were listening to DaBaby 98% of the time. Additionally, undergrad students are pregaming 29% of the time. Assume that 10. What is the probability that an undergrad is both listening to DaBaby and working out? a. 0.0483 b. 0.7681 c. 0.6958 d. 0.2842 e. None of the above C is the correct answer. Use conditional probability rule P(A | B) = (P(A intersect B)/P(B)) and the complement rule where P(A not) = 1 - P(A). Can solve it if needed. 11. Given an undergrad is listening to Ke$ha, what is the probability they are working out?

a. 02817 b. 0.0800 c. 0.1078 d. 0.8000 e. None of the above Use the following statistics for question 12-13: - 3% of college students are artists. - 80% of artists can solve a Rubik’s Cube. - 25% of college students that are not artists can solve a Rubik’s Cube. 12. What is the probability that a college student is an artist given that they know how to solve a Rubik’s cube? a. 0.024 b. 0.243 c. 0.09 d. 0.91 e. 0.757 C is the correct answer. Can be solved with Bayes’ rule or probability trees. T/F statements for question 15 i. Being an artist and not knowing how to solve a Rubik’s cube are collectively exhaustive ii. P(Can solve Rubik’s) + P(Artist) = P(Can Solve Rubik’s U Artist) iii. Given that a college student is an artist, we can assume she can solve a Rubik’s Cube. 13. Determine which of the above statements are true: a. only i b. i & ii c. ii & iii d. All of the above e. None of the above E is the correct answer. They are not mutually exclusive (not ii), nor completely exhaustive (not i) and only an 80% chance she can solve. 14. Given the following diagram, which of the following is P(A’ U B’)? [Assume numbers signify entire area]

Module 3 - Random Variables 16. 50 students are randomly selected from the undergraduate program. Let X be a binomial random variable that represents the number of in-state students. The overall probability of a student being in-state is 0.6. In-state students on average pay $16,000 for tuition, whereas out-of- state students on average pay $54,000 for tuition. What is the expected amount of tuition paid among the 50 randomly selected students? a. $2,700,000 b. $80,000 c. $1,940,000 d. $1,560,000 e. $1,750,000 D is the right answer. The expected value is calculated as 16,000*(0.6*50) + 54,000*(0.4*50) = 1,560,000. 17. Consider a store where purchases follow a poisson distribution. However the type of shopper does not follow a poisson distribution. There is a 40% chance that the shopper is an avid shopper as opposed to a casual shopper. Avid shoppers purchase 3.2 items per hour they spend in the store, and casual shoppers purchase 1.6 items per hour they spend in the store. Use the above information to fill in the blank: The probability of exactly 7 shoppers out of 10 being casual shoppers is ____ and if each of those 7 casual shoppers spend 1.5 hours each, the probability of less than 13 purchases from the casual shoppers is _____. a. 4.2% ; 14.5% b. 21.5% ; 21.4% c. 21.5% ; 14.5% d. 4.2%; 21.4% e. Not Enough Information (c) because we are looking for exactly 7 casual shoppers out of 10 with a probability of 0.7. This can be rewritten as =binom.dist(7 successes, 10 trials, 0.6 prob, 0 for exact) = 21.5%. As for the poisson, we are looking for P(X<13) which is P(X≤12). Our mean is 1.6 purchases/hour * 1.5 hours * 7 casual shoppers (independent). This can be rewritten as =POISSON.DIST(12 purchases, 16.8 expected, 1 for cumulative) = 14.5% 18. The number of miles walked by a person in Ann Arbor follows a Uniform Distribution 1, 8. What are the mean, standard deviation, and probability that someone walks exactly 7 miles?

a. μ = 4.5, = 0.29, P(x=7) = 0.86 b. μ = 4, = 2.02, P(x=7) = 0 c. μ = 4.5, = 2.02, P(x=7) = 0 ? d. μ = 4, = 0.29, P(x=7) = 0.86 e. μ = 4.5, = 2.02, P(x=7) = 0.86 ANSWER: C (probability of exact value in continuous distribution is 0) 19. Russell Westbrook’s free throw shooting for the 2018-19 NBA season followed a Normal Distribution with a mean of 65.6 and a standard deviation of 18.7 . What is the probability that Russ might shoot free throws between 5% below and 20% above last year’s average? Rounded to the nearest 2 decimal places . a. 11.31% b. 43.04% c. 75.85% d. 32.82% e. 34.09% ANSWER: D (norm.dist(1.2*65.6, 65.6, 18.7, 1) - norm.dist(0.95*65.6, 65.6, 18.7, 1)) 20. During the Fall Semester, students arrive at the Ross Starbucks following a Poisson distribution at an average rate of 24 customers per hour. What is the expected number of arrivals from 1pm to 4:15pm? a. 24 b. 48 c. 70 d. 78 e. None of the above. ANSWER: D (3.25*24 = 78) 21. During the Fall Semester, students arrive to the Ross Starbucks following a Poisson distribution at an average rate of 24 customers per hour. What is the probability that at least 100 customers arrive from 1:00pm – 5:00pm? (round to the nearest ten thousandth) a. 0.6818 b. 0.3182 c. 0.3549 d. 0.6451 e. None of the above. ANSWER: C, P(X>=100) = 1- P(X<=99) = 1- poisson.dist(99,96,1) -> 24*4 = 96 22. Kevin Malone enters the World Series of Poker. His probabilities of winning and potential

25. Apple Airlines overbooks its flights because it is known there is only a 90% chance a given ticket holder shows up for his or her flight. On a flight from Detroit to San Francisco, there are 100 seats but 102 ticket holders. Assume the probability one person shows up for a flight is independent of the probability another person shows up for the flight. What is the probability that everyone who shows up has a seat on this flight? Round to the nearest ten thousandth. a. 0.0003 b. 0.6667 c. 0.9997 d. 1.0000 n=102; p=0.9 P(X ≤ 100) = binom.dist(100, 102, 0.9, 1) = 0.9997 Use the following information for questions 28 and 29: The amount of time that Ross students spend on studying in a week is known to be normally distributed with a mean of 5.4 hours and a standard deviation of 1.6 hours. In order to incentivize students to study more, the Ross professors decide to honor students who have among the top 25% of study time of the whole student body in a dedicated ceremony next month. 26. You would like to be one of the students that are honored. How many hours at least would you need to study in a week to earn the opportunity? a. 4.5326 b. 6.4792 c. 7.4505 d. 8.2813 =NORM.INV(0.75, 5.4, 1.6) 27. You also hope that at least 5 other friends from your 8-person friend group (you are included as one of 8 people) can be honored in the ceremony with you. Assume that each person in your friend group have independent study time. What is the probability of your hope to come true? a. 0.1028 b. 0.0042 c. 0.9958 d. 0.8972 X~Bin(8, 0.25) P(X>=6) = 1-P(X<=5) = 1-BINOM.DIST(5, 8, 0.25, 1)