YOU HAVE FORM 1
TO 301: BUSINESS ANALYTICS AND STATISTICS
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TO 301 Midterm Exam Instructions
Before the exam:
1. Get out your student ID for the proctor to verify. If your attendance is not verified the
integrity of your exam may be questioned.
2. Turn off your phone. It must be put away the entire exam.
3. You may not use any wifi except for accessing Canvas during the exam.
4. Get out relevant notes and a calculator. You are not allowed to look in your bag once the
exam has begun.
a. Your calculator must be separate from your phone. Furthermore, it cannot
connect to the internet or external resources. The calculator cannot have wifi
capabilities at all.
b. You may use lecture notes, homework solutions, past exams, and excel templates;
you can also use all the files in Canvas.
5. Take off smart watches.
During the exam:
1. The exam has 30 questions and is worth 120 points. All questions are worth the same
number of points (4 points each).
2. Mark your answer on the scantron and also circle it in the exam booklet. Only the answers
you fill in on the scantron sheet will be graded. Each question has only 4 possible answers.
3. You may use the bathroom once, but you may not bring anything (including your phone)
with you. And you must sign in and out on the sheet at the front of the room. You will not
get extra time if you leave to use the bathroom.
4. You may leave the exam early until 30 minutes prior to stopping time. After that time,
you must remain in your seat until the end of the exam.
After the exam:
1. Stay seated. Separate your exam booklet and scantron. Wait until the proctor collects
both from all students.
2. Once ALL booklets and scantrons have been collected and counted, the proctors will
dismiss you.
2
Module 1
For Questions 1 to 2, please refer to the following:
A2S is a super store in Ann Arbor and it sells a variety of different products. The manager of A2S
is deciding which new products to carry in the store. Consider the following binary variables:
F = 1 if the manager decides to carry a new furniture piece
C = 1 if the manager decides to carry a new computer
T = 1 if the manager decides to carry a new TV-set
1. Which of the following constraints does F + C - T ≤ 1 model?
a. We must carry a new TV-set before we can carry both furniture piece and computer
together
b. We must carry a new TV-set before we can carry either a furniture piece or a
computer
c. We can only carry at most two products out of three: furniture piece, computer, and
TV-set
d. None of the above
2. Which of the following is appropriate to model “if we carry a furniture piece, we must carry
at least one product out of the two: either computer or TV-set”?
a. F ≥ C + T
b. F ≤ C + T
c. F + 1 = C + T
d. None of the above
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For Questions 3 to 5, please refer to the following figure:
y
E
D
F
A
C
B
x
G
The x and y axes in the above figure are equally scaled. The three dotted lines correspond to the
following constraints:
Const1: 𝑥 + 2𝑦 ≥ 4
Const2: 𝑦 ≥ 2
Const3: 2𝑥 − 𝑦 ≤ 8
They divide the whole region into 6 subregions: A, B, C, D, E, F, and G. Notice that there are no
constraints restricting negative values directly.
3. Which of the following is the feasible region for the above constraints?
a. A
b. B
c. C
d. E
4. Suppose that the objective is to minimize 2𝑦 − 𝑥. Which one is the optimal solution (if any)?
a. (x,y) = (0,2)
b. (x,y) = (4,0)
c. (x,y) = (5,2)
d. None of the above
5. Which of the following new constraints should be added to the above three constraints to
make the problem infeasible?
a. 𝑥 + 𝑦 ≤ 1
b. 𝑦 ≤ 1
c. 𝑥 ≤ 3
d. 𝑥 ≥ 3
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6. Solve the following LP using Excel Solver:
Minimize 3𝑥 − 2𝑦 − 𝑧
Subject to 𝑥 + 𝑦
≥2
𝑥
≥1
−2𝑥 + 5𝑦 + 𝑧 ≤ 10
𝑧≥0
𝑥 is an integer variable
𝑦 is a binary variable
What is the value of the objective function at the optimal solution?
a.
b.
c.
d.
0
-6
1
None of the above
Solution: x=2, y=0, z=14, objective function value=-8
For Questions 7 to 9, please refer to the following:
A furniture company manufactures tables and chairs. A table requires 40 board feet of wood, and
a chair requires 30 board feet of wood. Wood can be purchased at a cost of $1.50 per board foot,
and, at most, 40,000 board feet of wood are available for purchase. It takes 5 hours of skilled
labor to finish a table and 4 hours of skilled labor to finish a chair. A total of 5,000 hours of skilled
labor is available (and have already been paid for). A table can be sold at $220 per unit and a
chair can be sold at $175 per unit. Ignore integrality constraints (i.e., it is OK to have a fractional
solution). The company’s objective is to maximize total profits, and its decisions are (1) how many
board feet of wood to purchase, (2) how many tables to make, and (3) how many chairs to make.
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You should be able to answer Questions 7 to 9 using only the above sensitivity table.
7. What is the company’s total profit?
a. 218,750
b. 162,500
c. 56,250
d. None of the above
8. Suppose that the per unit revenue for chair changes from $175 to $174. Which of the
following statements is true?
a. Total profit increases by $1,250
b. The company will now make fewer chairs
c. Binding constraints do not change
d. None of the above
9. Suppose that the maximum number of skilled labor hours decreases to 4,000. Which of the
following statements is true?
a. Optimal solution does not change
b. Feasible region becomes larger
c. The LP becomes infeasible
d. Total profit changes
10. Which of the following statements is true?
a. An LP can have exactly two optimal solutions
b. The feasible region of an LP can consist of exactly one feasible point
c. An unbounded LP does not have an optimal solution
d. Everything else stay the same, it is not possible that maximizing and minimizing an
objective function yield exactly the same optimal solution
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Module 2
11. Let A and B be two events. If P(A) = 0.2, P(B) = 0.7, P(A ∩ B) = 0.1, then P(A ∪ B’) is equal to:
a. 0.2
b. 0.3
c. 0.4
d. 0.5
Solution:
Expand P(A ∪ B’) = P(A) + P(B’) – P(A ∩ B’), where
P(A) = 0.2,
P(B’) = 1 – P(B) = 1 – 0.7 = 0.3,
P(A ∩ B’) = P(A) – P(A ∩ B) = 0.2 – 0.1 = 0.1
Therefore, P(A ∪ B’) = 0.2 + 0.3 – 0.1 = 0.4.
12. On the Venn diagram in Figure 1, the shaded gray area represents:
a. (A ∪ B)’
b. (A ∩ B)’
c. (A ∩ B’) ∪ (A’ ∩ B)
d. None of the above
Figure 1:
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13. Let A and B be two events, such that P(A) > 0, P(B) > 0, and P(A | B) = 0. Which of the
following is TRUE?
a. A and B are mutually exclusive events
b. A and B are independent events
c. A and B are neither mutually exclusive nor independent events
d. A and B are both mutually exclusive and independent events
Solution:
Events A and B are mutually exclusive because P(A l B) = P(A ∩ B) / P(B) = 0, which implies
P(A ∩ B) = 0.
Events A and B are not independent because P(A | B) = 0 is not equal to P(A)>0.
14. Five hundred BBA students were surveyed about the difficulty of the subjects (TO,
Accounting, and Economics). Two hundred students thought TO is harder than Accounting,
three hundred students thought TO is harder than Economics, and fifty students thought TO
is easier than any other class. Which of the following statements is TRUE?
a. P (randomly chosen student thinks Econ is easiest of all) = 0.1
b. P (randomly chosen student thinks Econ is easiest of all) = 0.2
c. P (randomly chosen student thinks Econ is easiest of all) = 0.3
d. Not enough data to reject or accept any of the above statements
15. Client base of the prominent omni-channel retailer is such that 25% of its clients shop online
(i.e., they make purchases via online channel), while the rest shop offline (i.e., they make
purchases via offline channel by physically visiting retail stores). Based on the historical data,
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it is estimated that the probability that someone who visits the online channel will actually
make a purchase is P(purchase | online) = 0.7, while the probability that someone who
physically visits a store will actually make a purchase is P(purchase | offline) = 0.2. Find the
probability that a randomly sampled customer who happens to have made a purchase turns
out to be an online customer?
a. 0.18
b. 0.33
c. 0.54
d. 0.82
Solution:
P(online | purchase) = P(online ∩ purchase) / P(purchase), where
P(online ∩ purchase) = * P (online) * P (purchase| online) = 0.25*0.7 = 0.175.
P (purchase) = P (online)*P (purchase| online) + P (offline)*P (purchase| offline) =
0.25*0.7 + 0.75*0.2 = 0.325.
Therefore, P(online | purchase) = P(online ∩ purchase) / P(purchase) = 0.175 / 0.325 = 0.54.
16. The Ann Arbor car rental service is thinking of branching out to a new city (Lansing or Fort
Wayne). The table below provides the forecasted NPVs estimated for different realizations of
the state of the economy per branch. If the company doesn’t branch out at all, its NPV is
$150,000 for the High and Medium states of the economy, and $0 for the Low state. Which
of the following statements is TRUE?
State of the economy
High
Medium
Low
Branch out to Lansing
$700,000
$600,000
-$300,000
Branch out to Fort Wayne
$950,000
-$150,000
$250,000
a.
b.
c.
d.
Maximax and maximin decisions both prescribe branching out to Fort Wayne
Maximax and maximin decisions both prescribe branching out to Lansing
Maximin decision prescribes branching out to one city while maximax to the other
None of the above statements is true
Solution:
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Maximin decision prescribes not to branch out at all, as under “no branching out” worst scenario
is getting $0 which is better than worst scenarios of branching out to Lansing (-$300,000) or Fort
Wayne (-$150,000). Therefore, none of the a)-c) statements is correct.
17. Given the following statements:
1) Value of any amount of information cannot exceed EVPI,
2) Maximin decision coincides with Maximax decisions 50% of the time,
3) EMV maximizing decision never coincides with the Maximin decision,
4) It is always rational to make decisions to maximize EMV,
5) Paying for Perfect Information guarantees that the decision maker can choose the
outcome with the highest payoff,
6) Maximin and Maximax may correspond to the same decision.
How many of the statements (1)-(6) are TRUE?
a.
b.
c.
d.
1
2
3
More than 3
Solution:
Statements (1) and (6) are correct. Other statements are false.
18. A French bakery chain CroissantXL, is considering a merger with its competitor, bakery chain,
CroissantXS. The payoff that the CEO of CroissantXL is expected to receive will depend on the
next year’s market demand, which could be low or high. The payoff matrix for this decision
problem is shown in the table below. Based on the historical data, the probability of low and
high demand states is given by P(Low) = 0.4 and P(High) = 0.6. The CEO of CroissantXL and
the CEO of CroissantXS can jointly hire a market expert who could predict with 100% certainty
which state of the demand is to be realized next year. If they decide to hire the market expert,
CEOs will split the cost of the expert who charges $15,000.
Merge
Don’t Merge
Low Demand
$90,000
$40,000
High Demand
$130,000
$150,000
What is the minimum amount of money that the CEO of CroissantXS would have to pitch in
for the CEO of CroissantXL to be willing to agree to hire the expert?
a. $3,000
b. $4,000
c. $5,000
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d. $6,000
Solution:
The below calculations are in thousands of US dollars.
EMV = max {90*0.4 + 130*0.6, 40*0.4 + 150*0.6} = max(114,106}= $114
EMV with PI = 0.4*max{90,40} + 0.6*max{130,150} = $126
EVPI = EMV with PI – EMV = $126 - $114 = $12
The cost of the market forecast is $15, while EVPI of CroissantXL is 12$, therefore, the CEO
of CroissantXS should be able to cover at least $15-$12=$3 of the cost of the market demand
report.
19.
*** 19 is NOT included this year***
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Module 3
20. The following chart gives the cumulative distribution function of the market returns under
various scenarios of the future of the economy. Which of the following is the standard
deviation of the market return (choose the closest answer)?
a. 0.672
b. 0.058
c. 0.092
d. 0.984
Probability
0.10
0.20
0.25
0.35
0.10
Y
0.05
0.10
0.15
0.20
0.25
E[Y]
0.1575
(Y-E[Y])2
0.01156
0.00331
0.00006
0.00181
0.00856
V[Y]
0.00332
SD(Y)
0.05761
For Questions 21 to 22, please refer to the following:
Mountain Park ski store has stores at two locations in the Salt Lake City area: one in downtown
Salt Lake City (SLC) and one in Park City (PC). During the ski season, weekly ski sales at the SLC
store have a mean (expected value) of 20 pairs and a standard deviation of 5 pairs, while ski sales
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at the PC store have a mean (expected value) of 35 pairs and a standard deviation of 15 pairs.
The sales at the two locations are known to be independent. Profit margin from selling a pair of
skis at the SLC store is equal to $100, and profit margin from selling a pair of skis at the PC store
is $150.
21. What is the expected total weekly profit from the two stores (choose the closest answer)?
a. $5,250
b. $10,400
c. $7,250
d. $8,900
P=100X1 + 150X2. E(P)=100*E(X1) + 150*E(X2) = 100*20 + 150*35 = $7,250
22. What is the standard deviation of the total weekly profit from the two stores (choose the
closest answer)?
a. $4,143
b. $2,305
c. $1,152
d. $3,870
V(P) = 100^2*V(X1) + 150^2*V(X2) = 100^2*5^2 + 150^2*15^2 = 10,000*25 + 22,500*225 =
5,312,500
SD(P) = SQRT(5,312,500) = $2,305
For Questions 23 to 24, please refer to the following:
Alpha Air flies 50-passenger aircrafts between Detroit and NYC. Each ticketed passenger has a
10% chance of not showing up for the flight independent of other passengers.
23. If the airline sells 55 tickets for the flight, what is the probability that one or more of the
passengers will be bumped off the plane (choose the closest answer)?
a. 0.152
b. 0.015
c. 0.238
d. 0.345
If fewer than 5 passengers don’t show up (or equivalently more than 50 passengers
show up), one or more passengers will be bumped.
= binom.dist (4,55,0.10,1) =1-binom.dist (50,55,0.90,1) = 0.345
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24. If Alpha Air wants to keep the probability of bumping below 0.10, what is the maximum
number of tickets they should sell?
a. 51
b. 52
c. 53
d. 54
Trial and error:
Tickets
sold
51
52
53
54
Probability of
bumping
0.0046
0.0283
0.0898
0.1985
For Questions 25 to 26, please refer to the following:
Ale Pub in London plans to start happy-hour weekdays 4:00-6:00 pm during which they will sell
drinks at half-price. The manager, Adele, figured that Ale will break even if 22 customers arrive
during the two-hour period and make profits if more customers arrive. Adele promised to pay
an end-of-month bonus to each of its waiters if there are at least 15 profitable happy-hour days
within a month. The number of customers who arrive at the pub during happy hours follows
Poisson distribution with a mean rate of 12 customers per hour. Assume that there are 21 weekdays in a month and customer arrivals on different days are independent of each other.
25. What is the probability that Adele will pay bonuses in a given month (choose the closest
answer)?
a. 0.1227
b. 0.2227
c. 0.3227
d. 0.4227
Probability to have a profitable day =1-POISSON.DIST(22,24,1)=0.6083
Probability of having 15 or more profitable days =1-BINOM.DIST(14,21,0.6083,1)=0.2227
26. Suppose 17 customers arrived to Ale Pub during 4:00 to 5:30 pm on a Wednesday. What is
the probability that this day will end up being a profitable day (choose the closest answer)?
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a.
b.
c.
d.
0.3543
0.4543
0.5543
0.6543
If more than 5 customers arrive in the remaining 30 minutes, it will be a profitable
day:
Probability more than 5 customers in 30 minutes = 1-POISSON.DIST(5,6,1) = 0.5543
27. Gaël Monfils, a very athletic French tennis player, is known to play long matches against other
top professional tennis players. His matches in major tennis tournaments are known to last
anywhere from 3 to 5 hours and is equally likely to end at any time within this interval.
Furthermore, his chances of winning is 40% if the match duration does not exceed 4 hours,
and 70% if it does exceed 4 hours. Gaël is currently playing such a match and it has already
been three and a half hours. What is the probability that Gaël will win this match (choose the
closest answer)?
a. 0.50
b. 0.55
c. 0.60
d. 0.65
Let S: denote game is short (ends within 4 hours)
L: denote game is long (takes longer than 4 hours)
P(Win) = P(Win and S) + P(Win and L)
=P(Win|S)*P(S)+P(Win|L)P(L)
Given that the game lasted more than 3.5 hours, we can replace P(S) and P(L) with
revised probabilities: P(S|lasted longer than 3.5 hours)=1/3, P(L|lasted longer than
3.5 hours)=2/3
Hence P(Win) =(0.4)(1/3) + (0.7)(2/3) = 0.6.
For Questions 28 to 29, please refer to the following:
It is claimed that people are spending more time on Instantgram than Fatebook recently. A study
showed that the time a user spends on Instantgram daily has a mean of 30 minutes and a
standard deviation of 10 minutes, whereas the daily mean and standard deviation of time spent
on Fatebook are 27 minutes and 15 minutes, respectively. Both measures of time spent on
Fatebook and Instantgram are normally distributed and independent of each other.
28. What is the probability that a random Instantgram user spends longer time than a random
Fatebook user on a random day (choose the closest answer)?
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a.
b.
c.
d.
0.52
0.57
0.62
0.67
Let I and F be the time spent in Instantgram and Fatebook respectively.
Y = I-F is Normal with mean (30-27=3 minutes) and standard deviation, √102 + 152 ,
𝑁𝑜𝑟𝑚𝑎𝑙(3,18.03). P(Y>0) = 1-P(Y<0) =1-NORM.DIST(0,3,18.03,1) = 0.57.
29. What is the probability that an Instantgram user spends anywhere between 45 to 60
minutes on a random day on Instantgram (choose the closest answer)?
a. 0.025
b. 0.045
c. 0.065
d. 0.085
P(45 < I < 60) = P(I < 60) – P(I < 45) = NORM.DIST(60,30,10,1)-NORM.DIST(45,30,10,1) =
0.065
30. Based on a survey, Twinkler users spend, on average, 20 minutes tweeting and 30 minutes
reading the tweets of others daily. The tweeting time and the time reading tweets are both
normally distributed. It has been observed that 10% of the users spend a total time
(tweeting and reading tweets combined) of 80 minutes or longer daily. What is the
standard deviation of a random user’s total time spent on Twinkler (choose the closest
answer)?
a. 23.41
b. 15.54
c. 28.17
d. 31.26
Let T=T1+T2. E(T) = 20 + 30 = 50 minutes
P(T > 80) = 0.10 P(Z > (80-50)/sigma)=0.10. => z_0.90 =NORM.S.INV(0.9) = 1.28 =>
(80-50)/sigma=1.28 => Sigma=23.41
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