MGEB11H3–L01, L02
Quantitative Methods in Economics I
Test 2
Saturday July 16, 2022
13:00–15:00
Instructor: Victor Yu
Last Name (Print):
Sort Number
First Name (Print):
Student Number:
Aids allowed:
Any Calculator
Time allowed:
Two (2) hours
•
•
•
•
This test consists of 21 questions in 13 pages including this cover page. It is the student’s
responsibility to hand in a complete exam. Any missing page will get a zero mark.
Formula sheet and Statistical tables (Standard Normal) are provided separately.
Show your work in part 2. No marks will be given if you do not show your work.
This test is worth 30% of your course grade.
Turn to the last page and fill out your Name/Student Number again.
The University of Toronto's Code of Behaviour on Academic Matters applies to all University of Toronto Scarborough students. The
Code prohibits all forms of academic dishonesty including, but not limited to, cheating, plagiarism, and the use of unauthorized aids.
Students violating the Code may be subject to penalties up to and including suspension or expulsion from the University.
Page 1 of 13
Part 1 Multiple Choice. 3 points for each question. Total 51 points.
Select the BEST answer in each question. No part mark.
1. Suppose that W, X and Y are normal random variables with these means and variances:
Mean
Variance
W
2
4
X
0
9
Y
15
16
The smallest probability among P(W ≤ 1), P(X ≤ −2) and P(Y ≤ 12) is
(a) P(W ≤ 1)
(b) P(X ≤ −2)
(c) P(Y ≤ 12)
(d) not able to find from the given conditions
2. On the average, there are 4 customers entering a store in 10 minutes. Assume that the number of
customers entering the store independently and follow a Poisson distribution. In the next 30 minutes,
the probability that there are 4 customers entering the store every 10 minutes is closest to
(a) 0.00746
(b) 0.125
(c) 0.1954
(d) 0.25
(e) 0.5
3. If random variable X follows a normal distribution with mean 𝜇 and standard deviation 𝜎 , the
probability 𝑃(𝑋 ≥ 𝜇 + 1.5𝜎) is closest to
(a) 0.0668
(b) 0.3085
(c) 0.6915
(d) 0.8085
(e) 0.9332
4. A random sample of size 4 is selected from this normal population with mean 40 and standard
deviation 𝜎. Let 𝑋̅ be the sample mean. If 𝑃(20 ≤ 𝑋̅ ≤ 60) = 0.90, the value of 𝜎 is closest to
(a) 24.3161
(b) 12.158
(c) 21.053
(d) 22.222
(e) 44.444
Questions 5–6. The time to complete a task is a random variable X that follows a normal distribution
with mean µ and standard deviation of 10 minutes. Given that P(X > 10) = 0.7389.
5. The value of P(X > 20) is closest to
(a) 0.3594
(b) 0.1752
(c) 0.2142
(d) 0.4132
(e) 0.5018
6. If 300 observations are taken at random from this distribution, the number of observations greater
than 22 is closest to
(a) 86
(b) 100
(c) 112
(d) 121
(e) 130
Page 2 of 13
Questions 7–9. A random variable X has the following probability distribution
X
0
1
2
3
P(X)
0.2 0.3 0.3 0.2
7. If four observations on X are selected randomly from this distribution, the probability that one of them
takes on a value of 0 is equal to
(a) 0.1536
(b) 0.2
(c) 0.4096
(d) 0.8
(e) 0.8484
8. In a random sample of 100 observations from this distribution, the probability that the sum of these
X values is less that 180 is closest to
(a) 0.0256
(b) 0.1536
(c) 0.8
(d) 0.9744
(e) 0.9983
9. Let 𝑌 = 10 − 2𝑋 2 . The probability 𝑃(𝑌 < 0) is equal to
(a) 0.2
(b) 0.4
(c) 0.5
(d) 0.6
(e) none of these
Questions 10–12. It is reported that 30% of the population reads The Wall Street Journal, 20% read
the Financial Times, and 10% read both.
10.
If a person is randomly selected from this population, the probability that this person reads at least
one of the newspapers is
(a) 0.1
11.
(c) 0.3
(d) 0.4
(e) 0.5
If 10 people are randomly selected from this population, the probability that at least one of them
read both newspapers is
(a) (0.1)10
12.
(b) 0.2
(b) (0.9)10
(c) 1− (0.1)10
(d) 1− (0.9)10
(e) 1− 10(0.1)10
If 3 people are randomly selected from this population, what is the probability that only 2 of them
read at least one of the newspapers?
(a) 0.016
(b) 0.096
(c) 0.106
(d) 0.188
(e) 0.288
Page 3 of 13
Questions 13–17. Daily sales in a store follows approximately a normal distribution with mean $500
and standard deviation $200. Assume daily sales are independent from day to day.
13.
The probability that daily sales exceed $600 is closest to
(a) 0.1915
14.
(e) 0.5
(b) 670
(c) 710
(d) 756
(e) 820
(b) 0.6025
(c) 0.6433
(d) 0.695
(e) 0.7345
In the next 4 days, the probability that their average daily sales exceed $600 is closest to
(a) 0.1587
17.
(d) 0.495
Given that daily sales exceed $600, the probability that it also exceeds $650 is closest to
(a) 0.596
16.
(c) 0.45
The 90th percentile of daily sales is closest to
(a) 650
15.
(b) 0.3085
(b) 0.1915
(c) 0.3085
(d) 0.3413
(e) 0.445
In the next 4 days, the probability that the daily dales exceed $500 in only one of these days is
closest to
(a) 0.0313
(b) 0.05
(c) 0.095
(d) 0.125
(e) 0.25
Page 4 of 13
Part 2. Show your work in all questions. Total 49 points.
18.
(10 marks)
From previous years’ records, Air Canada knows that there is a 97% chance a passenger for the
Toronto to Montreal flight will show up, and assumes that passengers arrive independently of each
other. Each ticket for this flight costs $100. However, if a passenger shows up and cannot be
carried on the flight, Air Canada has to refund the $100 and pay a penalty of $400 to each such
passenger. How many tickets should Air Canada sell for a plane with 120 seats to maximize their
expected ticket revenues after paying for the possible refunds and penalty charges?
Page 5 of 13
19.
(10 points)
A sales company has an incentive program to compensate employees on the basis of their
performance. The company offers an annual bonus of $10,000 for superior performance, $6,000
for good performance, $3,000 for fair performance, and $0 for poor performance. Based on prior
records, an employee performs at superior, good, fair and poor levels with probabilities 0.1, 0.3,
0.4 and 0.2 respectively.
(a)
(5 points)
In a random sample of 210 employees, that is the probability that their average annual bonus is
over $4,500?
Page 6 of 13
19.
(b)
(continued)
(5 points)
In a large random sample of employees, the probability that their average annual bonus is below
$4,250 is 0.90. What is the size of this sample?
Page 7 of 13
20.
(a)
(10 points)
Two measurements X and Y are used to measure w, the weight of a new ship. Suppose X follows
a normal distribution with mean w and standard deviation 0.006w, Y follows a normal distribution
with mean w and standard deviation 0.004w.
(5 points)
Suppose that X and Y are independent, what is the probability that their average value is within
0.005w of the weight w?
Page 8 of 13
20.
(b)
(continued)
(5 points)
Suppose that X and Y are not independent, and their correlation coefficient is 0.2. What is the
probability that their average value is within 0.005w of the weight w?
Page 9 of 13
21.
(19 points)
Suppose that the prices for a litter of gas in Toronto and Vancouver are given as:
Average ($)
Standard Deviation ($)
Toronto
2.00
0.25
Vancouver
2.30
0.25
Assume that gas prices are independent among gas stations, and that gas prices follow a normal
distribution.
(a)
(3 points)
A random sample of 9 gas stations is selected from Toronto. What is the probability that the
average gas price in the sample is over $2.10?
(b)
(3 points)
A random sample of 5 gas stations is selected from Toronto. What is the probability that two of
them sell gas over $2.05 per litter?
Page 10 of 13
21.
(c)
(continued)
(3 points)
In a random sample of 500 gas stations from Toronto, what is the probability that more than 230
of them sell gas over $2.05 per litter?
(d)
(4 points)
A gas station is randomly selected from Toronto, and independently a gas station is selected from
Vancouver. What is the probability that the gas price in this Toronto station is less than that of
Vancouver?
Page 11 of 13
21.
(e)
(continued)
(6 points)
A random sample of 10 gas stations is selected from Toronto, and independently a random sample
of 10 gas stations is selected from Vancouver. What is the probability that the sample average
price in Toronto is less than that of Vancouver?
Page 12 of 13
MGEB11H3–L01, L02
Quantitative Methods in Economics I
Test 2
Saturday July 16, 2022
13:00–15:00
Instructor: Victor Yu
Last Name (Print):
Sort Number
First Name (Print):
Student Number:
Aids allowed:
Any Calculator
Time allowed:
Two (2) hours
•
•
•
•
This test consists of 21 questions in 13 pages including this cover page. It is the student’s
responsibility to hand in a complete exam. Any missing page will get a zero mark.
Formula sheet and Statistical tables (Standard Normal) are provided separately.
Show your work in part 2. No marks will be given if you do not show your work.
This test is worth 30% of your course grade.
Do not write on the space below, for markers only.
Page
2–4
5
6–7
8–9
10–12
Question
1–17
18
19ab
20ab
21abcde
Total
Max
51
10
10
10
19
100
Mark
Page 13 of 13