# 2012-07_Test_2S.pdf

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University of Toronto Scarborough

Economics for Management Studies

MGEB12H3

Daga

Winter

Description

ECMB12H3 Lec 01, 60
Quantitative Methods in Economics II
Department of Management
University of Toronto at Scarborough
Summer 2012
Dr. Yu
Test 2
Date: Saturday July 21, 2012
Time allowed: Two (2) hours
Aids allowed: Calculator and one aid sheet (two 8.5”x11” pages) prepared by student.
Notes:
• This test consists of 22 questions in 13 pages including this cover page.
• It is the student’s responsibility to hand in all pages of this test. Any missing page will get
zero mark.
• Statistical tables are provided separately. Do not hand in the tables. Only hand in your test
papers.
• Show your work in each question in Part 2.
• This test is worth 30% of your course grade.
Print Last Name Solution
Given Name(s)
Student Number
Circle your section L01 Wednesday L60 Online
Do not write on the space below, for markers only
Page Question Max Marks
2-7 1-18 54
8 19 10
9 20abc 12
10-11 21abc 12
12-13 22abc 12
Total 100
1 Part I. Multiple Choice. 3 points in each question. No part mark.
Circle only one answer. If there are more than one correct answer, circle the best one.
2
1. Let S be the variance of a random sample of siz e 25 from a population that is normal
with mean 3 and standard deviation 10. T he probabilitP 57.7018 < S <151.7292 ) is
closest to
(a) 0.1 (b) 0.3 (c) 0.5 (d) 0.7 √ (e) 0.9
2
Solution:P 57.7018 < S <151.7292 )= P57.7018*24 < n −1 S < 151.7292*24
10 2 σ 2 102
2 2
= P13.84843 < χ < 36.415 ) P13.84843 < χ < 36.415 )(0.95–0.05)=0.90
2. Two large random samples, each of size n, are selected independently from the same
population of mean µ and standard deviationσ . Let the sample means be X1 and X2 .
σ σ
The probabilityP − < X 1− X 2< is closest to
n n
(a) 0.2389 (b) 0.3455 (c) 0.4778 √ (d) 0.5222 (e) 0.7611
2
Solution: X is normal with meanµ and varianceσ ,X is normal with meanµ and
1 n 2
2
variance σ , and X ,X are independent. Therefore X − X is normal with mean 0
n 1 2 1 2
2 2 2
and varianceσ + σ = 2σ .
n n n
− σ −0 σ − 0
σ σ n n 1 1
P − < X1− X 2< = P 2 < Z < 2 = P− < Z <
n n 2σ 2σ 2 2
n n
= P − 0.7071< Z < 0.7071 0.5222
∧
3. If, from a sample, the simple regression line of Y on X is foundY =10−0.2X , and
∧
the simple regression line of X and YX = 5− 0.8Y . The sample correlation coefficient
is equal to
(a)− 0.16 √ (b)− 0.4 (c) − 0.6 (d) 0.16 (e)0.4
2 4. The time X for children over age 10 to complete a particular task follows a normal
distribution with mean 15 minutes and standard deviation 4 minutes. The time Y for
children at age 10 or under to complete the task follows a normal distribution with mean
18 minutes and standard deviation 3 minutes. Assume that X and Y are independent.
A random sample of 16 children over age 10 is selected and their average time to
complete the task is denoted by X . Independently a random sample of 16 children at
age 10 or under is also selected and their average time to complete the task is denoted by
Y . The probability P X >Y )is closest to
√ (a) 0.0082 (b) 0.3152 (c) 0.4918 (d) 0.6918 (e) 0.9918
Questions 5–7, to compare the proportion of people who support a certain government policy,
two independent random samples are selected from Ontario and Quebec, respectively.
The sample statistics are summarized below.
Sample Size Percent in Favour
Sample 1 from Ontario 400 62%
Sample 2 from Quebec 400 60%
Let p1 be the proportion of people from Ontario in favour of the policy, anp 2 be the
proportion of people from Quebec in favour of the policy.
5. To test if the proportions of people in support of the policy are equal between the two
provinces, the test statistic is closest to
(a) 0.05 (b) 0.1 0 (c) 0.25 √ (d) 0.55 (e) 0.90
6. At 5% significance level, which one of the following is the best conclusion?
√ (a) There is no significant difference betweep1 and p2.
∧ ∧ ∧ ∧
(b) There is no significant difference between 1 and p2 , where p1 and p2 are
point estimates ofp1 and p2, respectively.
(c) There is a significant difference betweep1 and p2.
∧ ∧ ∧ ∧
(d) There is a significant difference betweep and p , where p and p are point
1 2 1 2
estimates of p1 and p2 , respectively.
(e) None of the above
7. Which one of the following statements is true?
(a) A 90% confidence interval for 1 − p2 is wider than a 95% confidence interval.
(b) A 95% confidence interval for p − p must not contain the value 2%.
1 2
√ (c) A 95% confidence interval forp1− p 2 must contain the value 0%.
(d) A 95% confidence interval for p1− p 2must not contain the value 0%.
(e) None of the above is true
3 Questions 8–10, a real estate company wants to test whether the average sale price of residential
properties in North York is equal to the average sale price in Markham. The company
summarizes their sample data below:
City Sample Size Mean($000) Std Dev($000)
North York 40 340 48
Markham 40 325 50
Let µ 1and µ2 be the true average sale prices of residential properties in North York and
Markham, σ 1 and σ 2 be the true standard deviations of sale prices of residential
properties in North York and Markham, respectively.
8. A 95% confidence interval for µ1 -µ2 is closest to
(a)15±80.62 (b) 15± 62.73 √ (c)15 ± 21.48
(d) 15±18.03 (e) 15±12.65
9. Which one of the following statements is true in testing H 0 µ 1 µ 2 versus
H1 :µ1≠ µ 2 usingα = 0.05?
(a) There is a significant difference betweµ1 and µ2 .
√ (b) There is no significant difference between µand µ .
1 2
(c) Unable to conclude if there is a significant difference betµe1and µ2 .
(d) The p-value is smaller than 0.05.
(e) The probability of making a Type II error is 0.05.
10. Which one of the following statements is true?
(a) A 90% confidence interval for µ -µ is wider than the one in question 7.
1 2
(b) To test H0 : 1 = µ2 , the test statistit = 0.306.
(c) To testH : µ = µ , the test statistiZ = 0.306 .
0 1 2
(d) All of the above are true.
√ (e) none of the above is true.
4 Questions 11–12, a study if postoperative pain relief is conducted to determine if drug A has a
significantly longer duration of pain relief than drug B. Observations of the hours of pain
relief are recordedwith the following summary statistics.
Sample
Drug Size Mean (hours) Standard Deviation (hours)
A 10 4.6 1.5
B 10 4.0 1.8
11. To test if drug A has a significantly longer duration of pain relief than drug B, the value
of the test statistic is closest to
√ (a) t = 0.8098 (b) χ = 7.29 (c) χ = 8.1 (d) F =1.2 (e) F = 1.44
12. To test if the standard deviation on the duration of pain relief for drug B is significantly
less than 2 hours, the value of the test statistic is closest to
2 2
(a) t = 0.8098 √ (b) χ = 7.29 (c) χ = 8.1 (d) F=1.2 (e) F = 1.44
Questions 13–14, a random sample X ,X ,...,X is selected from a normal distribution with
1 2 9
mean µ and variance σ 2. Let X be the sample mean and S 2be the sample variance.
2 2
13. The probability P S > 2σ )is between
(a) 0 and 0.01 (b) 0.01 and 0.025 √ (c) 0.025 and 0.05
(d) 0.05 and 0.1 (e) 0.1 and 1
2 2 n − S) 2 (2 8 σ 2 2
Solution: P S > 2σ )= P 2 > 2 = P χ >16 ) which is between .025 and .05
σ σ
from the Chi-Square table with df = 8.
9
14. Suppose ∑ (X − X ) = 72 . A 95 percent confidence interval foσ 2is closest to
i= i
(a) (32.8496, 264.2544) √ (b) (4.1062, 33.0318) (c) (4.643, 33.032)
(d) (37.144, 264.256) (e) (16.354, 32.167)
(n− 1)S2 (n −1)S 2 72 72
Solution: 2 , 2 = , = (4.1062, 33.0318)
χ .025 χ .975 17.5345 2.17972
5 Questions 15-16, use the following information.
A random sample of 12 observations is us ed to estimate a simple regression relationship
between two variables. Here is a partial ANOVA table.
Source SS df MS F
Regression
Error 10
Total 350
15. What is the variance of the dependant variable (to the nearest integer)?
(a) 10 (b) 19 (c) 29 √(d) 32 (e) 35
16. What percent of the variation in the dependant variable is explained by variation in the
independent variable (to the nearest integer)?
√ (a) 71% (b) 76% (c) 82% (d) 92% (e) 95%
6 Questions 17–18, a simple regression analysis was per formed to predict Food Expend (food
expenditure) from Income based on a random sample of 7 households. The assumed
model is Food Expend = β0 + β1 Income + ε
Both income and food expenditure are measured in $100’s in the sample. A partial
computer printout is shown below.
Predictor Coef StdDev t-statistic P
Constant 1.8690 0.9068 2.06 0.094
Income 0.20195 0.04039 5.0 0.003
The scatter plot is given below.
17. Based on the sample data, the straight line relationship between income and food
expenditure is
(a) significant 0 < α < 0.001 .
(b) significant a0.001

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