VENUE: SEAT NUMBER: STUDENT NUMBER: FINAL EXAMINATION St Lucia Campus Second Semester, 2008 ECON1310 - Quantitative Economic & Business Analysis A PERUSAL TIME 10 mins. During perusal, write on the blank paper provided WRITING TIME 2 Hours EXAMINER Dr Averil Cook This examination paper has _12_ pages (include title page and attachments) and printed on Double-Sided THIS EXAMINATION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM Exam Type: Closed Book - Specified materials permitted Permitted Materials: Calculator - Yes - Casio FX82 Series with any letters Dictionary - No Other No electronic aids are permitted (e.g. laptops, phone) Formula sheet and statistical tables are attached to this paper. Answer: As instructed below Answer ALL questions in Part A in the answer booklet, and ALL questions in Part B on the MCQ sheet. Number of Questions: Part A 4; Part B 25. Weighting/Marks: 55% Special Instructions: Students must comply with the General Award Rules 1A.7 and 1A.8 which outline the responsibilities of students during an examination. ECON1310 - Final Semester Examination, Semester Two 2008 Page 2 of 15 PART A Answer ALL questions in Part A. Questions carry marks as indicated. A1. a. A flashlight battery is guaranteed to last for at least 40 hours. Tests indicate that the length of life of these batteries is normally distributed with a mean of 50 hours and a variance of 16 hrs sqd. What percentage of batteries will fail to meet the guarantee? (3 marks) b. A teacher gives a test to a large lecture class. It is known that the standard deviation of the scores is 10. If a random sample of 40 is selected from the class what is the probability that the sample mean will differ from the population mean by more than 4 marks? Do we need to be told that the class marks follow a normal distribution here? Explain. (3 marks) c. In a best-selling book the author claims that the average cost of a funeral is at least $7000. To test this claim an investigator examined a random sample of 20 funerals and found the mean to be $6425 with a standard deviation of $1080. Assume that funeral costs are normally distributed. Test the claim using o = 5%. Do we need to be told here that funeral costs are normally distributed? Explain. (7 marks) A2 a. An airline wants to determine the proportion of passengers that bring only carry-on luggage. How large a sample is required if the airline wants their estimate to be within 3% of the true proportion with 90% confidence? (2 marks) b. In a random sample of 200 customers, 44 have only carry-on luggage. Estimate the true proportion with 92% confidence. (3 marks) c. The airline considers that a new cheap service that allows only carry-on luggage would be profitable if the proportion of passengers currently in this category is over 20%. Test whether the proportion is more than 20% at the 5% level of significance. Use the p-value approach and the sample information in (b). Can they conclude that a new cheap service would be profitable? (5 marks) d. If it was later determined that the actual proportion was 30% what error, if any, was made? (2 marks) ECON1310 - Final Semester Examination, Semester Two 2008 Page 3 of 15 A3. A pharmaceutical company researcher wants to test whether habitual coffee drinkers (A), when given a cup of coffee just before bedtime, will take a shorter time to fall asleep compared to occasional coffee drinkers (B). A study of 16 habitual and 10 occasional coffee drinkers gave the following average times needed to fall asleep: Ax = 29 minutes; Bx = 56 minutes; As = 22 minutes; and Bs = 25 minutes. a. Compute the standard error for the difference in average times between the habitual and occasional coffee drinkers. (3 marks) b. Compute the 99% confidence interval for the difference in average times between the habitual and occasional coffee drinkers. Write an interpretation. From this confidence interval can we conclude that occasional coffee drinkers take longer to fall asleep than habitual coffee drinkers after drinking a cup of coffee just before bedtime? (3 marks) c. Test the hypothesis that drinking coffee before bedtime makes no difference to the length of time it takes to fall asleep for the two groups. Use = 0.01. Does the conclusion here agree with the conclusion in (b)? (5 marks) d. One of the assumptions associated with the calculations above is that the two samples are independent. Explain which calculation depends on this assumption. (2 marks) ECON1310 - Final Semester Examination, Semester Two 2008 Page 4 of 15 A4. An agent for a residential real estate company in a large city would like to be able to predict the monthly rental cost ($) for apartments based on the size of the apartment as defined by the number of square metres. A random sample of 25 apartments was selected and a simple linear regression equation was estimated. The printout is presented below. Regression Statistics Multiple R 0.75519 R Square Standard Error 202.4378 Observations 22 ANOVA df SS MS Regression 1 1087861 1087861 Residual 20 819620.9 40981.05 Total 21 1907482 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 259.26 215.28 1.20 0.24 -189.80 708.33 Size 9.89 1.92 5.15 0.00 5.88 13.89 Size Residual Plot-600-400-200020040060050 70 90 110 130 150 170SizeResiduals a. State the estimated simple linear regression equation for these data and explain the variables. (2 marks) b. Suppose you rented an apartment that was 5 sq metres larger than a basic size. Use this model to identify how much extra on the basic price you would expect to pay. (2 mark) c. Predict the price for an apartment of area 180 sq m. Does this involve extrapolation? Explain. (3 marks) d. What fraction of the variations in rent can be attributed to the size of the apartment? (2 marks) e. From the printout state the 95% confidence interval for the slope of the relationship in the population. Interpret. (3 marks) ECON1310 - Final Semester Examination, Semester Two 2008 Page 5 of 15 PART B MULTIPLE CHOICE QUESTIONS Answer ALL questions in Part B. B1. Consider the following statements: A. The t distribution is a family of normal distributions all with mean zero. B. The parameter for the t distribution is (n-1). C. For larger sample sizes the t distribution is less spread out. a. only A and B are true b. only A and C are true c. only B and C are true d. A, B and C are all true B2. When comparing the t distribution with the standard normal distribution a. they are both bell-shaped and normal b. they are both probability density functions and the probabilities for a particular value of t or Z can be found in the tables c. they both have the same mean but the t distribution has a smaller standard deviation than the Z distribution d. the t distribution has fatter tails than the Z distribution, and its standard deviation is more than 1 B3. The central limit theorem assures us that the sampling distribution of the mean a. is always normal b. approaches normality as the sample size increases c. none of the above d. is always normal when np and nq > 5 B4. An artist visits 36 homes selling his paintings. He considered he had a 20% chance of selling to any household. He managed to sell a painting to 10 of the households he visited. This sample proportion is part of the sampling distribution of the proportion of households (in a sample of 36) who buy, which a. is a normal distribution with ) p ( E = 0.20 b. is a binomial distribution with p = 0.20 c. is a normal distribution with ) p ( E =0.27 d. none of the above ECON1310 - Final Semester Examination, Semester Two 2008 Page 6 of 15 The next TWO questions use the following information A random sample of 60 garden centres was selected to determine the average number of fox-tail palms sold at the end-of-season clearance sales. A 95% confidence interval (5.62 to 6.38) was established based on the sample results. B5. The confidence interval would be narrower if a. the sample was larger b. the confidence level was larger c. the significance level was smaller d. all the above are correct B6. The confidence interval may be interpreted as a. If all possible samples of 60 are drawn from the population, 95% of the sample means will fall between 5.62 and 6.38 b. 95% of the garden centres have sold between 5.62 and 6.38 foxtail palms c. If all possible samples of 60 are taken from the population, 95% of the intervals developed will include the population mean. d. We are not totally certain (only 95%) that the true number of foxtail palms sold is between 5.62 and 6.38. B7. A market research company conducted a telephone survey of 765 households on behalf of a chain of new car dealers to determine the proportion of households seeking to purchase a new car in the next two years. An interval estimate of 0.076 to 0.112 was found. Determine the level of confidence that can be attached to this interval. a. 8.7% b. 95.6% c 91.3% d. 45.7% B8. A research report stated that the mean return on invested capital was between 5.8% and 10.8% per year with a confidence coefficient of 95%. Which of the following correctly interprets this interval estimate? a. 95% of investors will get investment returns between 5.8% and 10.8%. b. If many random samples were taken, 95% would have means between 5.8% and 10.8%. c. An investor expects to get a return between 5.8% and 10.8%. d. 95% of the population means will be between 5.8% and 10.8%. ECON1310 - Final Semester Examination, Semester Two 2008 Page 7 of 15 B9. When using small independent samples to estimate the difference between two population means the sample variances are pooled. This pooling process assumes a. the sample standard deviations are equal b. the sample variances are estimates of a common variance c. there is no relationship between the samples d. the populations are both normally distributed B10. When determining the sample size necessary to estimate a population proportion, if p is assumed to be 0.5 when in fact it does not equal 0.5, the value for n found will a. underestimate the sample size required b. overestimate the sample size required c. overestimate the sample size if the proportion is greater than 0.5 d. over- or under-estimate n depending on the true value of p B11. The local newspaper claims that no more than 5% of the residents of the community are on welfare. If you plan to test the claim by taking a random sample from the community, the appropriate alternative hypothesis would be a. p > 0.05 b. p < 0.05 c. p = 0.05 d. p s 0.05 B12. The approval process for selling life insurance is complex. The ability to deliver approved policies to customers in a timely manner is critical to the profitability of the firm. A random sample of 27 policies was selected and the processing time in days was recorded. To test whether the average processing time is better than 30 days the analyst should conduct. a. a two tail test b. a one tail test in the right tail c. a one tail test in the left tail d. no test - just find the sample average time B13. Which of the following statements is TRUE in a hypothesis test? a. The larger the level of confidence, the more likely you are to reject H0. b. The level of significance is the probability of a Type I error. c. The alternative hypothesis cannot ever be proven to be true since it is always a very general statement. d. The significance level is the probability of a Type II error. ECON1310 - Final Semester Examination, Semester Two 2008 Page 8 of 15 B14. Which of the following is TRUE? a. The alternative hypothesis represents the conclusion for which evidence is sought. b. The statement of the null hypothesis never contains an equality. c. An increase in the risk of a Type I error also increases the risk of a Type II error. d. The computed test statistic is also known as 'the critical value'. B15. The Queensland government is doing a survey on the popularity of daylight savings. If more than 70% of the population are in favour then it will be introduced as a permanent feature. Which of the following are the correct hypotheses to test? a. H0: p s 0.7 Ha: p > 0.7 b. H0: p = 0.7 Ha: p < 0.7 c. H0: p = 0.7 Ha: p = 0.7 d. H0: p > 0.7 Ha: p s 0.7 B16. A manufacturer of car batteries claims that his product will last at least 4 years on average. A sample of 50 is taken and the mean and standard deviation are found. The test statistic is calculated to be -1.656. Using a 5% significance level, the conclusion would be: a. There is insufficient evidence for the manufacturer's claim to be considered correct. b. There is insufficient evidence for the manufacturer's claim to be considered incorrect. c. There is sufficient evidence for the manufacturer's claim to be considered correct. d. There is sufficient evidence for the manufacturer's claim to be considered incorrect. B17. In the past 32% of all materials shipments to a large manufacturing company were received late. The company believes that their new just-in-time system in which suppliers are linked more closely to the manufacturing process, will be an improvement. For the hypothesis test conducted, the critical value was -1.75 and the calculated value was -1.86. The Null hypothesis would _________ and the company would conclude that their new system is _________. a. not be rejected; successful b. not be rejected; unsuccessful c. be rejected; successful d. be rejected; unsuccessful ECON1310 - Final Semester Examination, Semester Two 2008 Page 9 of 15 B18. Which of the following is TRUE? If a ________ null hypothesis is _______ a Type II error has been made. a. true, rejected b. true, not rejected c. false, rejected d. false, not rejected B19. Given the null hypothesis H0: s 350, and the decision rule Reject H0 if Zcalc > 1.85, a type II error will occur when a. = 360, Zcalc = 2.01 b. = 360, Zcalc = 1.67 c. = 340, Zcalc = 2.01 d. = 340, Zcalc = 1.67 B20. If a Null hypothesis is not rejected at the 5% level of significance it a. will never be rejected at the 1% level b. will always be rejected at the 1% level c. will sometimes be rejected at the 1% level d. there is insufficient information to say what will happen at the 1% level B21. The least squares criterion is used a. to calculate the coefficient of determination b. to calculate the covariance between two variables c. to obtain the scatter plot of the data d. to obtain the sample regression constant and coefficient B22. Suppose the estimated linear relationship between the circumference (C) of pumpkins (cms) and their weight (kgs) was found to be W = -2.6 + 0.08C. The observed weight for a circumference of 51cms was 2.23kg. The calculated residual is a. 750 gm b. 1.72 kg c. 0.37 kg d. 2.42 kg B23. It is the belief that there is a positive relationship between productivity growth and employment growth. To test this belief data from twelve OECD countries were obtained. The critical t value for testing this belief at the 1% level of significance would be a. 2.764 b. 3.106 c. 3.169 d. 2.718 ECON1310 - Final Semester Examination, Semester Two 2008 Page 10 of 15 B24. The Kaddstat output below is for a regression to estimate the number of goals scored in AFL based on the number of kicks. Regression and Correlation Observations 16 R Square 0.1039 Standard Error 1.0899 Multiple R 0.3223 Coefficients Standard Error t value p value Intercept 5.1253 6.9380 0.7387 0.4723 Kicks 0.0460 0.0361 1.2738 0.2235 The standard deviation of the points around the estimated regression line is calculated to be a. 0.0361 b. 6.9380 c. 0.3223 d. 1.0899 Econ1310 final exam 2008 sem 2 ANSWERS Q1.a) variable X = length of life (hours) = 50 hrs, o2 = 16 o = 4 P(X < 40) = =|.|

\| 7000 (claim = at least) Ha: < 7000 dec rule: reject H0 if tcalc < -t.05(19) = -1.729 tcalc = == 2010807000 6425nsx-2.38 reject H0 Sufficient evidence at the 5% level of significance, to conclude that funeral costs are less than $7000 on average. YES we do need to be told that funeral costs are normally distributed because the test statistic t was used (because only the sample standard deviation is known) and the sample size is small. (7 marks) Q2.a) 67 . 75103 . 05 . 0 645 . 1Epq zn22 222>> > At least 752 people are required in the sample for the sampling error to be no more than 3% (2 marks) b) 22 . 020044p = = 92% confidence interval for p 20078 . 0 22 . 075 . 1 22 . 0nq pz p04 . 0 = ] 0292916 . 0 [p = o = 0.22 0.05126 0.168 < p < 0.271 true proportion of passengers who only bring carry-on luggage is estimated, with 92% confidence, to be between 17% and 27% (3 marks) c) H0: p s 20%= 0.2 Ha: p > 0.2 (over 20%) dec rule: reject H0 if p-value < o = 0.05 zcalc = 0282842 . 002 . 02008 . 0 2 . 02 . 0 22 . 0npqp p== = 0.707 p-value = P(Z > 0.71) = 0.5 0.2611 = 0.2389 (not < 0.05) Do not reject H0 Insufficient evidence at the 5% level of significance to conclude the proportion of passengers who take only carry-on luggage is more than 20%. Cannot conclude the new cheap service would be profitable. (5 marks) d) if real proportion was 30% then the H0 statement is false a False H0 was not rejected so a Type II error was made. (2 marks) if students wrongly use 0.02929 for the st error the Z value would be 0.6828 (incorrect) ECON1310 - Final Semester Examination, Semester Two 2008 Page 12 of 15 Q3.a) 875 . 53624128852 10 1625 ) 1 10 ( 22 ) 1 16 (s iance var pooled2 22p = = + + = = |.|

\| + =|.|

\| + =101161875 . 536n1n1s difference for error st2 12p = 87.242 = 9.34 (3 marks) b) 99% confidence interval for A B = B x A x ) 24 ( 005 . 0 B As t x x 29 56 2.797 9.34 = 27 26.124 53.12 < A B < 0.88 or 0.88 < B A < 53.12 The average time taken to fall asleep for habitual coffee drinkers (A) is estimated with 99% confidence to be between 0.88 and 53.12 minutes less than for occasional coffee drinkers. Yes - we can conclude that on average occasional coffee drinkers take longer to fall asleep than habitual coffee drinkers after drinking a cup of coffee just before bedtime because the upper and lower values of the confidence interval both have the same sign which indicates that B is larger than A (3 marks) c) H0: B A = 0 Ha: B A = 0 dec rule: Reject H0 if |tcalc | > t0.005(24) =2.797 tcalc = 34 . 90 27S) ( ) X X (B x A xB B A B = = 2.89 Reject H0 Sufficient evidence at the 1% level of significance to conclude that there is a difference in the average time taken for habitual and occasional coffee drinkers to fall asleep after drinking a cup of coffee at night. YES this test shows rejection in the right tail, so we can surmise that the average time for B (occasional ) is greater than for A (habitual). The confidence interval also showed that the average time for B was always greater than for A so same conclusion. (5 marks) d) The calculation of the st error for the difference does not include any covariance so is only relevant if the samples are independent. (2 marks) Q4.a. X 89 . 9 26 . 259 Y + = where X = Size of the apartment (sq m) Y = monthly rental cost ($) (2 marks) b) Since coeff 9.89 indicates the increase in rental cost for a one sq m increase in size if X = 5 expected extra price would be 5* 9.89 = $49.45 (2 marks) c) Y= 259.26 + 9.89 180 = 2039.46 ie monthly rental predicted to be $2039.46 Since the maximum value of X on the residual plot is about 150, the value 180 lies outside the range in the sample so YES it is an example of Extrapolation (3 marks) ECON1310 - Final Semester Examination, Semester Two 2008 Page 13 of 15 d) r2 = 5703 . 019074821087861SSTSSR= = 57% of variations in rent can be attributed to the size of the apartment (2 marks) e) 95% confidence interval for 1 from printout is 5.88 < 1 < 13.89 [or calculated from 1 b ) 20 ( 025 . 0 1S t b = 9.89 2.086 1.92] Interpretation: The slope of the linear relationship in the population between the size of apartments and the monthly rental cost is estimated, with 95% confidence, to be between 5.88 and 13.89. (2 marks) PART B 1. C 2. D 3. B 4. A 5. A 6. C 7. C 8. no correct answer 9. B 10. B 11. A 12. C 13. B 14. A 15. A 16. B 17. C 18. D 19. B 20. A 21. D 22. A 23. A 24. D 25. D ECON1310 - Final Semester Examination, Semester Two 2008 Page 14 of 15 B25. The following residual plot was obtained from a regression relating demand to the difference in prices between two competitors' products. PriceDif Residual Plot-0.6-0.4-0.200.20.40.60.81-0.20 0.00 0.20 0.40 0.60 0.80PriceDifResiduals The plot indicates that a. no assumptions have been violated b. the assumption of independence between the error terms has been violated c. a problem of homoscedasticity exists d. the assumption of constant error variance has been violated END OF PAPER ECON1310 - Final Semester Examination, Semester Two 2008 Page 15 of 15 Econ1310 Semester 2 2008 FORMULA SHEET NORMAL AND SAMPLING DISTRIBUTIONS for the appropriate distribution: test statistic (Z or t) = error dard tan sparameterpopulationstatisticsample STANDARD ERRORS for (i) Distribution of the (ii) Distribution of the sample mean x sample proportion p n=xoo npq =p o (iii) Distribution of difference between means (with pooling) 2 - n + n1)s - (n + 1)s - (n = s here wn1 +n1s = s2 122 221 1 2p2 12p x x2 1 ||.|

\| ESTIMATION and HYPOTHESIS TESTING Confidence interval estimate = errordard tan sstatisticteststatisticsample Determining sample size: mean 2Ezn2||.|

\| o> o ; proportion n > 22Epq z2o; SIMPLE LINEAR REGRESSION Coefficient xxxy1SSSS = b constant x b y = b1 0 SSxx = n) x (x = ) x x (22 2 (similarly for SSyy) ) y y )( x x ( = SSxy SST = SSyy = SSR + SSE SSR = xx21SS b Standard errors: 2 nSSE= se SSs = Sxxeb1 Coefficient of Determination : yy yy2SSSSE1SSSSR = r =