Test lasts for 120 minutes. You must stay for the entire 120 minute period.

Transcription

1 ECO220 Mid-Term Test (June 29, 2005) Page 1 of 15 Last Name: First Name: Student ID #: INSTRUCTIONS: DO NOT OPEN THIS EAM UNTIL INSTRUCTED TO. Test lasts for 120 minutes. You must stay for the entire 120 minute period. You may have 2 pens with black or blue ink, a non-programmable nongraphical calculator, and your student ID card out during the exam. Nothing else is permitted. Formulas and statistical tables are given in a separate handout. True/False and Multiple Choice : You must write final answer on the line provided. For these questions you do not need to show work. No partial credit. Problems : Significant points will be awarded only to responses that are clear, complete, and precise. Failure to show work or explain the answer will result in a 0 mark. If you run out of room you may continue your answer on page 15, but indicate you have done so and clearly label your additional responses (for example: Question (2) (b) continued: ). True/False : 10 points (10 questions * 1 point each) Multiple Choice : 45 points (15 questions * 3 points each) Problems : 45 points (3 questions with varying point values) Total possible: 100 points

2 ECO220 Mid-Term Test (June 29, 2005) Page 2 of 15 True/False (10 points total): Write T if true or F if false on the line provided. Each worth 1 point. (1) Adding up the heights of the bars in a frequency histogram yields the sample size. (2) In general, as the sample size increases you should reduce the number of bins in a density histogram. (3) The standard error of the population mean is given by: σ. N (4) The interquartile range is robust to outliers. (5) A 90% confidence interval estimator is more likely to exclude the true population parameter with a sample size of 10 compared to a sample size of 100. (6) Inferences should not be based on very small samples because estimators from such samples are biased. (7) If is a normal random variable then random variable. Y = 2 * 10 e is also a normal (8) If 1, 2, 3, 4, 5 are independent random variables with identical probability distributions, then 5 ] E is equivalent to ] [ 1 E [ 5 i. i= 1 (9) If 1, 2, 3, 4, 5 are independent random variables with identical probability distributions, then [ 51] V [ 5 i. i= 1 V is equivalent to ] (10) If a purely random sample is selected and the sample does not suffer from non-response or selection bias, then you can conclude that you have experimental data and not observational data.

3 ECO220 Mid-Term Test (June 29, 2005) Page 3 of 15 Multiple Choice Questions (45 points total): Write the letter of the best answer on the line provided. Each worth 3 points. (1) In which case is it reasonable to expect that the sample median will be greater than the sample mean? (a) Sample with n = 10 taken from a positively skewed population (b) Sample with n = 1,000 taken from a negatively skewed population (c) Sample with n = 2 taken from a positively skewed population (d) Sample with n = 100 taken from a uniform population (2) The sample standard deviation and the coefficient of variation of the data shown in this histogram is closest to which of the following? (a) s.d. 0.4 and cv 0.02 (b) s.d. 1.0 and cv 0.05 (c) s.d. 2.0 and cv 0.10 (d) s.d. 2.0 and cv 0.20 (e) s.d. 4.0 and cv 0.20 Density (3) Suppose a firm believes that there is a linear and causal relationship between the number of promotions it runs and total sales of its good. The mean number of promotions is 200 per month with a standard deviation of 20. Sales on average are 1,000 per month with a standard deviation of 200. The covariance between promotions and sales is 10,000. How many extra units should the firm expect to sell if it runs 10 additional promotions? (a) 10 (b) 15 (c) 20 (d) 25

4 ECO220 Mid-Term Test (June 29, 2005) Page 4 of 15 (4) Airlines often place their passengers in tiers. Depending on frequency of travel and dollars spent, a passenger could be: super elite, elite, member, or regular. Suppose that only 2% of an airline s passengers are super elite. If the baggage of all passenger tiers is handled in the same way, what is the probability (to the nearest hundredth) that of the 10 people who had their bags lost by the airline, none of them are super elite? (a) 0.82 (b) 0.90 (c) 0.91 (d) 0.98 (5) Consider a population with a variance of 100. To obtain a 95% interval estimator of the population mean such that the difference between the upper and lower confidence limits is no more than 2 units would require a sample size of at least: (a) 20 (b) 271 (c) 385 (d) 38,416 (6) Consider the data summarized in the tabulation below. Which of the following is a reasonable inference? num_trips Freq. Percent Cum Total (a) Sample was taken from a positively skewed log-normal population (b) Sample was taken from a Normal population with µ 1.5 (c) Sample was taken from a Poisson population with λ 0.6 (d) Sample was taken from a Binomial population with n = 3 and p 0.5 (e) Sample was taken from a Uniform population with a = 0 and b = 3

5 ECO220 Mid-Term Test (June 29, 2005) Page 5 of 15 For questions (7) (10) consider the following information: The human resources (HR) department of a company with 10,000 employees conducted a survey to assess employee expectations related to compensation. A random sample of 100 employees reported their anticipated annual bonus () in dollars and their anticipated annual salary increase (Y) in dollars. Union rules prevent negative bonuses and salary decreases. HR calculated the following: = 3,000 s = 2,000 Y s Y = 2,000 = 1,000 r = 0.50 (7) Which of the following statements is most plausible? (a) Both and Y are normal (b) Both and Y are approximately normal (c) is approximately normal but Y is not normal (d) Neither or Y is normal (e) Insufficient information provided to make any assessment of distributions (8) Based on these survey results what is the point estimate of the total dollar amount (includes bonuses and salary increases) that the company should set aside if it wants to meet its employees expectations? (a) $50,000 (b) $500,000 (c) $5 million (d) $500 million (9) Which of the following scatter diagrams is plausible when and Y are measured in 1000 s of dollars: (a) Scatter Diagram #1 (b) Scatter Diagram #2 (c) Scatter Diagram #3 (d) All of the above (e) Either Scatter Diagram #2 or #3 Y Scatter Diagram # Y Scatter Diagram # Y Scatter Diagram #

6 ECO220 Mid-Term Test (June 29, 2005) Page 6 of 15 (10) What is the s.d. of the amount of additional compensation (includes bonus and salary increase) expected by employees? (a) $55 (b) $1,732 (c) $2,236 (d) $2,646 (e) $3,000 (11) Suppose a manufacturing company has determined its total cost function, which shows the relationship between output (Q) and total costs (TC). Based on engineering reports it finds that the total cost function has the following functional form: TC = f + m*q, where f is fixed costs and m is marginal (variable) costs. The company knows its fixed costs are $100 million dollars and its marginal costs are $2.50. If on average Q is 1 million units, then what is the expected value of total costs (TC)? (a) $2.5 million (b) $102.5 million (c) $100.0 billion (d) $250.0 billion (12) For which of these populations would a sample size of 10,000 yield the narrowest (most precise) 95% confidence interval estimator of the population mean? (a) Population #1 (b) Population #2 (c) Population #3 (d) Populations #1 and #3 are equally narrow (and more narrow than Population #2) (e) Insufficient information provided to make this determination Population #1 mean: 40, sd: Density Population #2 mean: 2.3, sd: Population #3 mean: -20, sd:

7 ECO220 Mid-Term Test (June 29, 2005) Page 7 of 15 (13) Consider the sample of data shown in this graph. The 95% confidence interval estimator (rounded to nearest tenth) of µ is: (a) LCL = 8.5 and UCL = 11.5 (b) LCL = 9.0 and UCL = 11.0 (c) LCL = 9.5 and UCL = 10.5 (d) LCL = 9.9 and UCL = 10.1 Density n: 100, mean: 10.0, sd: (14) Suppose and Y are independent. ~ U[0, 10] and Y ~ U[-10, 0], where U[a, b] denotes the uniform distribution and its parameters. If a sample of 100 observations of each is taken, how is the sum of the sample means distributed? (a) U[-10, 10] (b) U[-1, 1] (c) U[-0.1, 0.1] (d) T[-10, 10], where T[2a, 2b] denotes the Triangle distribution (15) Suppose that on an average Saturday night during peak season, 24 of a hotel s 304 rooms request room service. What is the probability that on a given Saturday night during peak season 35 or more rooms request room service? (a) 0.01 (b) 0.02 (c) 0.03 (d) 0.04 (e) 0.05

8 ECO220 Mid-Term Test (June 29, 2005) Page 8 of 15 Problems (45 points total): Show your work/explain. (1) [10 pts] A real estate agency has only two sales agents: Diane and Dan. Each sales agent makes either 1 or 2 sales a month. The broker knows the joint probability distribution of sales for the two agents but wants to know the standard deviation of total sales. Find it. Show work and give final answer in a sentence and indicate units of measurement. Diane Dan

9 ECO220 Mid-Term Test (June 29, 2005) Page 9 of 15 (2) [18 pts] Suppose a market researcher is assigned to study how many caffeinated soft drinks (like Coke, Pepsi, etc.) that a typical high school student purchases in a day from vending machines on school property. (a) [5 pts] What is the relevant population? How is it distributed? Support any claims you make. (Be sure to discuss whether it is a continuous or discrete distribution.)

10 ECO220 Mid-Term Test (June 29, 2005) Page 10 of 15 (b) [2 pts] What population parameters are of interest given the assignment and what are the plausible estimators? For (c) and (d): Researcher s sampling plan: Interviewer stands next to a particular vending machine at a particular high school and randomly selects a sample of 150 students that purchase a caffeinated soft drink from the machine. Each of the randomly selected students is asked: How many soft drinks did you drink today? (c) [8 pts] What are four major problems with this sampling plan and survey given the market researcher s original assignment? Explain each.

11 ECO220 Mid-Term Test (June 29, 2005) Page 11 of 15 Answer for (c) continued: (d) [3 pts] Would the proposed estimator(s) be unbiased? Consistent? Explain and indicate any directions of bias.

12 ECO220 Mid-Term Test (June 29, 2005) Page 12 of 15 (3) [17 pts] Consider a normal population with µ = 7 and σ 2 = 27: ~ N(7, 27). (a) [2 pts] What is the probability of obtaining a negative value of? (b) [3 pts] What is the probability of getting a negative sample mean if n = 30?

13 ECO220 Mid-Term Test (June 29, 2005) Page 13 of 15 For parts (c) and (d): Consider a population that is uniform: Y ~ U[ 2, 16]. (c) [3 pts] What is the probability of obtaining a negative value of Y? (d) [3 pts] What is the probability of getting a negative sample mean if n = 30?

14 ECO220 Mid-Term Test (June 29, 2005) Page 14 of 15 (e) [6 pts] Compare your answers in (a) and (c). Compare your answers in (b) and (d). Explain why you get the same or different answers.

15 ECO220 Mid-Term Test (June 29, 2005) Page 15 of 15 Extra Space: If you use this space, indicate for which question(s).