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THE UNIVERSITY OF MELBOURNE DEPARTMENT OF ECONOMICS SEMESTER 1 ASSESSMENT, 2013 ECON10005 QUANTITATIVE METHODS 1 Reading Time: 15 minutes Writing Time: 2 hours

This examination paper contributes 70 percent to the assessment in ECON10005 Quantitative Methods 1. The following items are authorised in the exam room: - Foreign language/English dictionaries. - Non-programmable calculators may be used.

This document has 15 pages including the formula sheet, statistical tables and the sample response sheet. This paper is not to be removed from the exam room. This paper will not be held in the Baillieu Library. The examination is comprised of two parts: PART A: MULTIPLE CHOICE. ANSWER ALL 10 QUESTIONS. This part contributes 30 marks to this examination. Suggested time allocation: 50 Minutes. Answer all questions. Colour in the small circle in the appropriate space with a 2B pencil on the RESPONSE SHEET. Please follow the SAMPLE RESPONSE SHEET (see page 9 of this document) for details required on the formal RESPONSE SHEET. Each question is worth 3 marks. An incorrect answer, no answer, or more than one answer will all receive a zero mark.

PART B: PROBLEMS. ANSWER ALL QUESTIONS. This part contributes 40 marks to this examination. Suggested time allocation: 70 Minutes. Question 1 is worth 7 marks, Question 2 is worth 10 marks, Question 3 is worth 12 marks, Question 4 is worth 11 marks. Answer all questions in the examination booklet(s) provided. Page 1 of 15

PART A: MULTIPLE CHOICE ANSWER ALL QUESTIONS Question 1 (3 marks) Suppose in a one-tailed hypothesis test where the alternative hypothesis is that the population parameter is less than a specified value, the test statistic is z = 1.5. The p-value will be: (a) 0.0334 (b) 0.0668 (c) 0.4666 (d) 0.9332 (e) None of the above Question 2 (3 marks) In a mechanic’s workshop, the probability that a customer complains about the price charged on any given day is 20%. The probability of a customer complaining on one day is independent of whether or not a customer complained on any other day. What is the probability that, in a random sample of 4 days, a customer complains about the price on less than 2 days? (a) 0.0016 (b) 0.0256 (c) 0.1536 (d) 0.4096 (e) None of the above Question 3 (3 marks) A company wants to estimate the proportion of employees that are sick on a randomly selected day. What sample size is required if the company wishes to be at least 99% confident that their estimate will be correct to within 0.05? (a) 271 (b) 385 (c) 542 (d) 664 (e) None of the above

Page 2 of 15

Question 4 (3 marks) In regression analysis, which of the following terms describes the variation in y that is not explained by variation in x? (a) Σ(y − yˆ)2 (b) Σ(ˆ y − y)2 (c) Σ(y − y)2 (d) R2 (e) None of the above Question 5 (3 marks) If you wanted to compare the amount of variation in one set of data with the amount of variation in another set of data, where the two sets of data were measuring different variables in different units, the most appropriate statistic to use would be: (a) the standardised z score for a randomly selected observation from each set of data (b) the standard error of the estimate from a linear regression using one set of data as the y variable and the other as the x variable (c) the coefficients of variation from each set of data (d) the correlation coefficients for the two sets of data (e) None of the above Question 6 (3 marks) If we estimate a regression model of the form yi = β0 + β1 xi + ǫi , (i = 1, 2, . . . , n), using ordinary least squares, then which of the following statements is true? (a) The sum of the residuals will be zero (b) The mean of the residuals will be zero (c) The sum of the squared residuals may not be zero, but will be minimised (d) All of the above (e) None of the above

Page 3 of 15

Question 7 (3 marks) Compared to simple random sampling, cluster sampling is: (a) likely to be cheaper, and likely to be more accurate (b) likely to be cheaper, but risks being less accurate (c) likely to be more expensive, but likely to be more accurate (d) likely to be more expensive, and likely to be less accurate (e) None of the above Question 8 (3 marks) The central limit theorem states that: (a) the sampling distribution of the sample mean X will be approximately normally distributed (even if X is not normally distributed) so long as the sample size is sufficiently large (b) the sampling distribution of the sample mean X will be approximately normally distributed (even if X is not normally distributed) so long as the number of samples is large (c) the sampling distribution of the sample mean X will be approximately normally distributed if X is normally distributed, regardless of the sample size (d) All of the above (e) None of the above Question 9 (3 marks) If L is a normally distributed random variable with a mean of 47 and a standard deviation of 3, and L0.05 is the value of L with 0.05 beyond it in the right tail of the distribution, then which of the following pairs of statements is true? (a) L0.05 = 1.645, and P (µL ≤ L ≤ L0.05 ) = 0.95 (b) L0.05 = 1.645, and P (µL ≤ L ≤ L0.05 ) = 0.45 (c) L0.05 = 51.935, and P (µL ≤ L ≤ L0.05 ) = 0.95 (d) L0.05 = 51.935, and P (µL ≤ L ≤ L0.05 ) = 0.45 (e) None of the above

Page 4 of 15

Question 10 (3 marks) An industry report on caf´es states that the strength of a cup of coffee (measured by the number of beans ground to make the cup of coffee) is normally distributed, with a mean of 20 beans per cup. You take a random sample of 81 cups, and observe a mean strength of 18.4 beans per cup with a sample standard deviation of 9 beans per cup. If you test the hypothesis that the average cup of coffee weaker than reported at the 5% level of significance, which of the following statements is true? (a) The test statistic will be −1.6, and we will reject the null (b) The test statistic will be −1.6, and we will not reject the null (c) The test statistic will be 1.6, and we will reject the null (d) The test statistic will be 1.6, and we will not reject the null (e) None of the above

Page 5 of 15

PART B: PROBLEMS ANSWER ALL QUESTIONS Question 1 (7 marks) (a) Explain why correlation is a better measure of the linear association between two variables than covariance. (2 marks) (b) Consider the following statement: Because correlation is better than covariance as a measure of the linear relationship between two variables we never need to calculate covariance. Is this statement true or false? Explain your answer.

(1 mark)

(c) If X is a random variable, prove that the correlation of X with itself is equal to 1, both for a sample of size n drawn from the population of all possible X values, and also for a population of X values of size N . (4 marks) Question 2 (10 marks) Let X1 , . . . , Xn denote a simple random sample from a normally distributed population, with mean µ and known variance σ 2 , represented by the random variable X, i.e. X ∼ N(µ, σ 2 ). On the basis of this sample, consider two possible interval estimators for µ that might be used: Estimator 1: Xi ± zα/2 σ

1 σ Estimator 2: X ± zα/2 √ , where X = n n

(1) n X

Xi .

(2)

i=1

where Xi denotes a single observation from the sample (the ith observation), and zα/2 satisfies the equation P (Z > zα/2 ) = α/2, where Z is a normally distributed random variable with a mean of 0 and a standard deviation of 1. (a) Using the probability statement P (−zα/2 ≤ Z ≤ zα/2 ) = (1 − α) as a starting point, derive the interval estimator in equation (1). (4 marks) (b) Which of the two estimators should be used in practice? Why?

(2 marks)

(c) Explain why, if X is not normally distributed, it is inappropriate to use the estimator defined in equation (1). (2 marks) (d) Under what condition can we use the estimator defined in equation (2) even if X is not normally distributed? (2 marks)

Page 6 of 15

Question 3 (12 marks) You read a report that states that in a sample of 200 new businesses, 15% failed within 1 year, 40% failed in 1–2 years, and the remainder continued for at least 5 years. The same report notes that half of owners of businesses that failed within 1 year were starting their first business, half of the owners of businesses that failed in 1–2 years were starting their first business, and just one third of owners of businesses that continued for at least 5 years were starting their first business. (a) Estimate the probability that a randomly selected new business continues for at least 5 years. (1 mark) (b) Estimate the probability that a randomly selected new business fails within 1 year, given that the owner of the business is not starting their first company. (1 mark) (c) Based on your answers to parts (a) and (b), do you believe that the longevity of a business is independent of whether or not the owner has previously started a business? Justify your answer. (1 mark) (d) Test, at the 1% level of significance, the proposition that the longevity of a business is independent of whether or not the owner has previously started a business. (7 marks) (e) Do your answers to parts (c) and (d) contradict each other? Explain why or why not. (2 marks) Question 4 (11 marks) Let GDP denote the annual percentage change in a country’s gross domestic product, MIG denote the annual percentage change in the number of migrants arriving in a country, FDI denote the annual percentage change in foreign direct investment in a country, and POP denote the annual percentage change in the population of a country. Further, define the variable OIL as an indicator of whether or not a country is an oil producer; specifically OI L = 1 if a country is an oil-producing country and OI L = 0 otherwise. Using a sample of observations on 16 countries, a regression model for GDP in terms of the other variables is estimated using Excel and the output is presented in Table 1 on page 8. (a) State and interpret the estimated coefficient on FDI.

(3 marks)

(b) Based on these estimates, is the elasticity of GDP with respect to the number of migrants arriving in a country elastic, unit elastic or inelastic? Explain why. (2 marks) (c) Interpret the estimated coefficient on OI L × F DI.

(3 marks)

(d) Interpret the estimated coefficient on OI L.

(3 marks)

END OF QUESTIONS Page 7 of 15

Table 1: Excel Output of Regression Results for Question 4

Page 8 of 15

Page 9 of 15

Table 1: Cumulative Standardised Normal Probabilities*

z

0

P(1 < Z < z) 0.00

Z

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

3.0

0.0013

0.0013

0.0013

0.0012

0.0012

0.0011

0.0011

0.0011

0.0010

0.0010

2.9

0.0019

0.0018

0.0018

0.0017

0.0016

0.0016

0.0015

0.0015

0.0014

0.0014

2.8

0.0026

0.0025

0.0024

0.0023

0.0023

0.0022

0.0021

0.0021

0.0020

0.0019

2.7

0.0035

0.0034

0.0033

0.0032

0.0031

0.0030

0.0029

0.0028

0.0027

0.0026

2.6

0.0047

0.0045

0.0044

0.0043

0.0041

0.0040

0.0039

0.0038

0.0037

0.0036

2.5

0.0062

0.0060

0.0059

0.0057

0.0055

0.0054

0.0052

0.0051

0.0049

0.0048

2.4

0.0082

0.0080

0.0078

0.0075

0.0073

0.0071

0.0069

0.0068

0.0066

0.0064

2.3

0.0107

0.0104

0.0102

0.0099

0.0096

0.0094

0.0091

0.0089

0.0087

0.0084

2.2

0.0139

0.0136

0.0132

0.0129

0.0125

0.0122

0.0119

0.0116

0.0113

0.0110

2.1

0.0179

0.0174

0.0170

0.0166

0.0162

0.0158

0.0154

0.0150

0.0146

0.0143

2.0

0.0228

0.0222

0.0217

0.0212

0.0207

0.0202

0.0197

0.0192

0.0188

0.0183

1.9

0.0287

0.0281

0.0274

0.0268

0.0262

0.0256

0.0250

0.0244

0.0239

0.0233

1.8

0.0359

0.0351

0.0344

0.0336

0.0329

0.0322

0.0314

0.0307

0.0301

0.0294

1.7

0.0446

0.0436

0.0427

0.0418

0.0409

0.0401

0.0392

0.0384

0.0375

0.0367

1.6

0.0548

0.0537

0.0526

0.0516

0.0505

0.0495

0.0485

0.0475

0.0465

0.0455

1.5

0.0668

0.0655

0.0643

0.0630

0.0618

0.0606

0.0594

0.0582

0.0571

0.0559

1.4

0.0808

0.0793

0.0778

0.0764

0.0749

0.0735

0.0721

0.0708

0.0694

0.0681

1.3

0.0968

0.0951

0.0934

0.0918

0.0901

0.0885

0.0869

0.0853

0.0838

0.0823

1.2

0.1151

0.1131

0.1112

0.1093

0.1075

0.1056

0.1038

0.1020

0.1003

0.0985

1.1

0.1357

0.1335

0.1314

0.1292

0.1271

0.1251

0.1230

0.1210

0.1190

0.1170

1.0

0.1587

0.1562

0.1539

0.1515

0.1492

0.1469

0.1446

0.1423

0.1401

0.1379

0.9

0.1841

0.1814

0.1788

0.1762

0.1736

0.1711

0.1685

0.1660

0.1635

0.1611

0.8

0.2119

0.2090

0.2061

0.2033

0.2005

0.1977

0.1949

0.1922

0.1894

0.1867

0.7

0.2420

0.2389

0.2358

0.2327

0.2296

0.2266

0.2236

0.2206

0.2177

0.2148

0.6

0.2743

0.2709

0.2676

0.2643

0.2611

0.2578

0.2546

0.2514

0.2483

0.2451

0.5

0.3085

0.3050

0.3015

0.2981

0.2946

0.2912

0.2877

0.2843

0.2810

0.2776

0.4

0.3446

0.3409

0.3372

0.3336

0.3300

0.3264

0.3228

0.3192

0.3156

0.3121

0.3

0.3821

0.3783

0.3745

0.3707

0.3669

0.3632

0.3594

0.3557

0.3520

0.3483

0.2

0.4207

0.4168

0.4129

0.4090

0.4052

0.4013

0.3974

0.3936

0.3897

0.3859

0.1

0.4602

0.4562

0.4522

0.4483

0.4443

0.4404

0.4364

0.4325

0.4286

0.4247

0.0

0.5000

0.4960

0.4920

0.4880

0.4840

0.4801

0.4761

0.4721

0.4681

0.4641

*

Source: E. A. Selvanathan, S. Selvanathan, G Keller (2011). Business Statistics - Australia and New Zealand, 5th Edition,

Cengage Learning Australia Pty Limited.

Page 10 of 15

Table 1 (continued): Cumulative Standardised Normal Probabilities*

0

z

P(1 < Z < z) Z

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

0.5000

0.5040

0.5080

0.5120

0.5160

0.5199

0.5239

0.5279

0.5319

0.5359

0.1

0.5398

0.5438

0.5478

0.5517

0.5557

0.5596

0.5636

0.5675

0.5714

0.5753

0.2

0.5793

0.5832

0.5871

0.5910

0.5948

0.5987

0.6026

0.6064

0.6103

0.6141

0.3

0.6179

0.6217

0.6255

0.6293

0.6331

0.6368

0.6406

0.6443

0.6480

0.6517

0.4

0.6554

0.6591

0.6628

0.6664

0.6700

0.6736

0.6772

0.6808

0.6844

0.6879

0.5

0.6915

0.6950

0.6985

0.7019

0.7054

0.7088

0.7123

0.7157

0.7190

0.7224

0.6

0.7257

0.7291

0.7324

0.7357

0.7389

0.7422

0.7454

0.7486

0.7517

0.7549

0.7

0.7580

0.7611

0.7642

0.7673

0.7704

0.7734

0.7764

0.7794

0.7823

0.7852

0.8

0.7881

0.7910

0.7939

0.7967

0.7995

0.8023

0.8051

0.8078

0.8106

0.8133

0.9

0.8159

0.8186

0.8212

0.8238

0.8264

0.8289<...