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MGEB12H3 (26)
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Midterm

# 2013-03_Test2S.pdf

12 Pages
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Department
Economics for Management Studies
Course
MGEB12H3
Professor
Daga
Semester
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) = PZ > 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 PX 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 ,PZ > 1  = 0.2,  n   2σ 2/n   2σ    1 = 0.84, σ = 0.8417938 2σ
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