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Economics for Management Studies

MGEB12H3

Daga

Winter

Description

ECMB12H3S – LEC01, L30, L60
Quantitative Methods in Economics II
Department of Management
Winter 2013
Test 2
Date Friday March 22, 2013; 3:00–5:00pm
Time allowed Two (2) hours
Aids allowed Calculator and one aid sheet (two 8.5”x11” pages) prepared by
student.
Instructor Victor Yu
Instructions
• This test consists of 22 questions in 12 pages including the cover page.
• It is the student’s responsibility to hand in all pages of this tesAny missing
page will get zero mar2.
• Statistical tables (Zχ , F) and a regression formula sheet are provided separately.
• Show your work in each question in Part 2.
• This test is worth 30% of your course grade.
Last Name (Print) Solution
First Name (Print)
Student Number
Circle your section L01 Tuesday 12-3 L30 Wednesday 7-10
L60 Online
Do not write on the space below, for markers only
Page Question Max Marks
2-6 1–17 51
7 18ab 10
8 19ab 10
9 20ab 9
10-11 21abc 10
11-12 22ab 10
Total 100
Management, 1265 Military Trail, Toronto, ON, M1C 1A4, Canada
www.utsc.utoronto.ca/mgmt
1 Part I. Multiple Choice. 3 marks in each question. No part mark.
Circle only one answer. If there are more than one correct answer, circle the bestone.
1. A simple regression model is assumed for relating the price of grapefrui(in
dollars) to quantities of grapefruit demanded Y . Data for five months is provided
below and some calculations are given.
2
X Y X − X (X − X ) Y −Y X − X Y −Y )
0.1 50 -0.2 0.04 22 -4.4
0.2 30 -0.1 0.01 2 -0.2
0.3 30 0 0 2 0
0.4 20 0.1 0.01 -8 -0.8
0.5 10 0.2 0.04 -18 -3.6
Total 1.5 140 0 0.10 0 -9.0
Mean 0.3 28 0 0.02 0 -1.8
The slope of the estimated simple regression line is equal to
√ (a) –90 (b) 90 (c) –55 (d) 55 (e) none of these
2. A sample of size 100 selected from one population has 60 successes, and a sample
of size 150 selected from a second population has 95 successes. The test statistic
for testing the equality of the population proportions is closest to
√ (a) –0.5319 (b) 0.7293 (c) –0.419 (d) 0.2702 (e) 0.5
3. If, from a sample, the simple regression line of Y on X is found to be
Y =10 − 4 X , and the simple regression line of Xand Y is X = 5− 1Y . The
3 3
sample correlation coefficient is
(a)− 1 √ (b) − 2 (c) 1 (d) 2 (e) none of these
3 3 3 3
4. Let X ,X ,...,X and Y ,Y ,...,Y are independent random samples from a
1 2 9 1 2 16
normal population with standard deviation 1. Denote the sample means by X
and Y , respectively. The probabiliP X −Y >1 )is closest to
√ (a) 0.0082 (b) 0.1915 (c) 0.3085 (d) 0.4918 (e) 0.5
2 Questions 5 and 6. Use the following paragraph.
Let X1 be the mean of a random sample of size 100 from a population with mean
20 and standard deviation 12;X 2 be the mean of a random sample of size 100
from a population with mean 20 and standard deviation 5. Assume thatX1 and
X 2 are independent.
5. Which one of the following follows approximately a standard normal
distribution?
1 1 10
(A) (X 1 X 2) (B) X 1 X 2) √(C) (X1− X 2 )
13 17 13
(D) 10 X − X ) (E) none of these
17 1 2
6. The probabilityP X 1 X 2 2)is closest to
√(A) 0.0618 (B) 0.1673 (C) 0.25 (D) 0.3751 (E) 0.4382
Questions 7-9. Use the following information.
A telemarketing company obtains customer names and telephone numbers from
two sources. To determine if the percentages of customers who make purchases
are the same from both sources, random sample A is selected from one source and
independently random sample B is selected from the other source. Both samples
are large samples. Sample A shows that 20% of customers make purchases and
sample B shows only a 18%.
7. A 90% confidence interval for the difference in the percentages of customers who
make purchases from sources A and B is (0.005, 0.035). If there are 4000
customers in sample A, the number of customers in sample B is closest to
(a) 3012 √ (b) 3421 (c) 4000 (d) 4520 (e) 4900
8. Suppose sample A and sample B consist of 4000 customers each, a 90 per cent
confidence interval for the difference in the percentages of customers who make
purchases is closest to
(a) 0.0± 0.0275 (b) 0.02±0.0214 (c) 0.0± 0.0195
(d) 0.0± 0.0172 √ (e) 0.0± 0.0144
9. Suppose sample A and sample B consist of 4000 customers each. To test if the
percentages of customers who make purchases are the same from both sources,
the p-value is closest to
(a) 0.005 (b) 0.0113 √ (c) 0.0226 (d) 0.0312 (e) 0.05
3 Questions 10-12. Use the following information.
A simple regression analysis is performed to relate maintenance expense ( Y,
dollars per month) to usage ( X, hours per week). A random sample of 82
observations showed an F-statistic of 10 in the ANOVA table, and the scatter plot
suggested that maintenance expense increases as usage increases. The sample
variance of Y is 100.
10. The sample correlation coefficient between Xand Y is closest to
1 1 1 1 1
(a) (b) (c) √ (d) (e)
9 5 4 3 2
11. The value of MSE (Mean Squared Error) in the ANOVA table is closest to
(a) 60 (b) 70 (c) 80 √ (d) 90 (e) 100
1
12. If the sample variance ofX is , the slope of the simple regression line is closest
36
to
(a) 1 (b) 5 (c) 10 (d) 15 √ (e) 20
Questions 13-15. Use the following information.
A market research firm published the following summary statistics for the
population of university students regarding their working status in the years of
2001 and 2002.
In 2001, 50% of the students worked. In 2002, 52% of the students
worked. Among the students who worked, the following population statistic s
are given:
Weekly Income
Year Mean Std Dev
2001 $190 $50
2002 $200 $50
13. A random sample of 400 students (no matter they worked or not) is selected from
year 2001, and independently a random sample of 400 students (no matter they
worked or not) is selected from year 2002. What is the probability that the
percent of students working from the sample of 2001 is higher than the percent of
students working from the sample in 2002?
(a) 0.001 (b) 0.002 (c) 0.003 (d) 0.004 √ (e) 0.005
∧
Solution: p ~ N p ,p1 1 = N 0.5,(0.5 0.5 = N .5, 0.25 ,
1 1 n1 400 400
p ~ N p ,p2 2 = N 0.52,(0.52 0.48 )= N 0.52, 0.2496
2 2 n 400 400
2
4 ∧ ∧ p1 1 p2 2 0.25+0.2496 0.4996
p1− p2 ~ N 1− p 2, n + n = N − 0.02, 400 = N− 0.02, 400
1 2
∧ ∧ ∧ ∧ 0− −0.02 )
P 1 > p2 = P 1− p2 > 0= P > = P Z > 0.5659 = 0.2843
0.4996/400
14. A random sample of 100 students is selected from the students who worked in
2001, and independently a random sample of 100 students is selected from the
students who worked in 2002. What is the probability that the average weekly
income from the sample of 2001 is higher than the average weekly income from
the sample in 2002?
(a) 0.01 (b) 0.02 (c) 0.04 (d) 0.06 √ (e) 0.08
502 502
Solution: X1 ~ N 190, = N 190,25), X 2 ~ N 200, = N 200,25 )
100 100
X1− X 2 N −10,50 ),
0 − − 10)
P X 1 X >2 0)= P > = P Z > 1.4142) = 0.0793
50
15. Suppose weekly income for working students follow approximately a normal
distribution in each year. A student who worked in 2001 is randomly selected,
and independently a student who worked in 2002 is also selected. What is the
probability that the student from 2001 has a higher weekly income than the
student from 2002?
(a) 0.0557 (b) 0.0795 (c) 0.2833 (d) 0.3722 √ (e) 0.4443
Solution: X1~ N 190,50 2 , X 2 N 200,50 2),X 1 X 2 ~ N − 10,5000)
P X − X > 0) = PZ > 0− − 10) = P Z > 0.14142 )= 0.4443
1 2 5000
5 Question 16-17. Use the following information.
Two large random samples, each sample size is n, are selected independently
from the same population of mean µ and standard deviationσ . Let the sample
means be X1 and X2 .
16. The probabilityP σ < X − X < σ is closest to
n 1 2 n
(a) 0.1587 (b) 0.2611 √ (c) 0.5222 (d) 0.8418 (e) 0.95
σ2
Solution: X1 is normal with mean µ and variance , X 2 is normal with meanµ and
n
σ 2
variance , and X1 ,X2 are independent. Therefore X1− X 2 is normal with
n
σ 2 σ 2 2σ 2
mean 0 and variance + = .
n n n
σ σ − σ / n −0 σ / n − 0
P < X1− X 2 = P < Z <
n n 2σ 2/n 2σ 2 /n
= P − 1 < Z < 1 =P − 0.71< Z < 0.71 = 0.5222
2 2
1
17. Suppose PX 1 X >2 n = 0.2. The value ofσ is closest to
(a) 0.1587 (b) 0.2611 (c) 0.5222 √ (d) 0.8418 (e) 0.95
Soln:P −1X > 2 1 = 0.2,P Z > 1/ n −0 = 0.2 ,PZ > 1 = 0.2,
n 2σ 2/n 2σ
1 = 0.84, σ = 0.8417938
2σ

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