Eco 251 Formula List

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ECO 251 QBA1 Name FINAL EXAM, Version 1 Class ______MAY 8, 2006

Part I. Do all the Following (14 Points) Make Diagrams! Show your work! Illegible and poorly presented sections will be penalized. Exam is normed on 75 points. There are actually 123+ possible points. If you haven’t done it lately, take a fast look at ECO 251 - Things That You Should Never Do on a Statistics Exam (or Anywhere Else).

x ~ N13,5.6

1. P0  x  28

2. F12.00 (Cumulative Probability)

3. Px  28

4. P28  x  32

5. P3  x  3

6. x.23 (Find z.23 first)

7. A symmetrical region around the mean with a probability of 23%. [14, 14]

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II. (10 points+, 2 point penalty for not trying part b.) Show your work! Mark individual sections clearly.

Wynn, Anthony and Avronovic give us the following data for a sample of eight professional golfers. Ea or x is earnings in thousands of dollars and SA or y is average score.

Row Ea x SA y 1 71.6 71.50 2 55.8 72.75 3 147.4 71.34 4 117.4 71.27 5 112.3 70.95 6 82.7 71.65 7 22.8 72.49 8 58.6 71.46 In order to speed things up, I have computed the sum of the first seven observations.

7 7 7 7 7 2 2  x  610.000,  y  501.950,  x  63720.1,  y  35996.0 and  xy  43607.4. i1 i1 i1 i1 i1

Calculate the following: a. The sums that you will need to calculate whatever parts you do. (1 point if you don’t quit at b) Make sure that I can tell how you did these sums. b. The sample standard deviation s y of average score. (2)

c. The sample covariance s xy between x and y . (2)

d. The sample correlation rxy between x and y . (2) e. Given the size and sign of the correlation, what conclusion might you draw on the relation between x and y ? (1) Can you guess why the correlation isn’t stronger? f. Assume that the earnings of the golfers were 15% lower ( w  .85x ). Find w (the sample mean 2 s r of earnings), sw , wy and wy . Use only the values you computed in a-d and rules for functions of

x and y to get your results. If you state the results without explaining why, or change x1 and x2 and recompute the results, you will receive no credit. (4). g. Do an 80% confidence interval for the population average score of professional golfers. (2) [14, 28]

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III. Do at least 5 of the following 6 sections (at least 12 each) (or do items adding to at least 48 points - Anything extra you do helps, and grades wrap around) . Show your work! Please indicate clearly what sections of the problem you are answering! If you are following a rule like Eax  aEx please state it! If you are using a formula, state it! If you answer a 'yes' or 'no' question, explain why! If you are using the Poisson or Binomial table, state things like n , p or the mean. Avoid crossing out answers that you think are inappropriate - you might get partial credit. Choose the problems that you do carefully – most of us are unlikely to be able to do more than half of the entire possible credit in this section!) This is not an opinion questionnaire. Answers without reasons or supporting calculations or table references will not be accepted (except in multiple choice questions)!!!! Answers that are hard to follow will be penalized. Note that some sections extend over more than one page.

A. Answer the following 6 multiple choice questions. (These should be 2 each, but to discourage guessing, how about 2.5 each for right answers and 0.5 penalty for wrong answers.)

1. The t distribution should be used when the parent (underlying) population a) Is Normal, the population standard deviation is unknown and we are testing a mean. b) Is Normal, the population standard deviation is known and we are testing a mean. c) Is Normal, the mean of the population is unknown and we are testing a mean. d) Is binomial and we are testing for a proportion. e) The t distribution should be used in all of these cases.

2. (Was 5) It is desired to estimate the average total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval was calculated to be ($2,181,260, $5,836,180). Which of the following interpretations is correct? a) 97% of the sampled total compensation values fell between $2,181,260 and $5,836,180. b) We are 97% confident that the mean of the sampled CEOs falls in the interval $2,181,260 to $5,836,180. c) In the population of Service industry CEOs, 97% of them will have total compensations that fall in the interval $2,181,260 to $5,836,180. d) We are 97% confident that the average total compensation of all CEOs in the Service industry falls in the interval $2,181,260 to $5,836,180.

3. (Was 9) In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a interval. a) narrower b) wider c) less significant d) biased

4. (Was 13) For air travelers, one of the biggest complaints is of the waiting time between when the airplane taxis away from the terminal until the flight takes off. This waiting time is known to have a skewed-right distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights. a) Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutes. b) Distribution is skewed-right with mean = 10 minutes and standard error = 8 minutes. c) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8 minutes. d) Distribution is approximately normal with mean = 10 minutes and standard error = 8 minutes.

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5. (Was 16) Why is the Central Limit Theorem so important to the study of sampling distributions? a) It allows us to disregard the size of the sample selected when the population is not normal. b) It allows us to disregard the shape of the sampling distribution when the size of the population is large. c) It allows us to disregard the size of the population we are sampling from. d) It allows us to disregard the shape of the (parent) population when n is large.

6. (Was 21) What type of probability distribution will the consulting firm most likely employ to analyze the insurance claims in the following problem? An insurance company has called a consulting firm to determine if the company has an unusually high number of false insurance claims. It is known that the industry proportion for false claims is 3%. The consulting firm has decided to randomly and independently sample 100 of the company’s insurance claims. They believe the number of these 100 that are false will yield the information the company desires. a) binomial distribution. b) geometric distribution c) continuous uniform distribution d) Poisson distribution. e) hypergeometric distribution. f) none of the above. [15]

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B. A random sample of 36 Kleenex users taken at a college found that the average sneezer used 52.10 tissues in the course of a cold. The researcher previously believed that the average was 59. Do a 95% confidence interval for the population mean and answer if the population mean is significantly different from 59.00 for parts a-d. Note that each number here is stated to the nearest hundredth. Please maintain at least that level of significance throughout the problem. Mark your individual questions clearly!!! a. Assume that the population standard deviation is known to be 20.95. (4) b. Assume, more realistically, that 20.95 was a sample standard deviation. (4) c. Now assume that the sample of 36 was drawn from a population of 401. (4) d. Assume that the population standard deviation is known to be 20.95 but that we want a confidence level of 96% (You cannot use the t-table for this part.).(4) e. Assume that the researcher was right and that the average number of Kleenex used by a sneezer has a population mean of 59 and a population standard deviation of 20.95. Assume that the researcher has 2250 tissues on hand for the 36 subjects. What is the probability that the researcher runs out of tissues? (4) [35]

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C. Assume that PA  .25 and PB  .70 . (Making joint probability tables will help. – do not round excessively, you should retain at least 3 figures to the right of the decimal point.) 1. Find (i) PA B , (ii) PA  B(iii) PB Aif a. A and B are independent. (3) b. A and B are mutually exclusive. (3) c. PA B  .20 (3)

2. If PA  .40 and PB  .70 , show that A and B cannot be mutually exclusive. (3)

3. At a Pennsylvania state college, 60% of freshmen come from Eastern Pennsylvania, 30% from Western Pennsylvania, and 10% from out of state. They are given a math test during freshman week. 70% of the students from Eastern Pennsylvania pass it as do 60% from from Western PA and 90% from out of state. Tara Bulsnob comes from Eastern PA and has passed the math test and will only be friends with someone else from Eastern PA who has passed the math test.. a. If she picks someone at random, what is the chance that that person has passed the math test.? (2) b. If she picks someone who has passed the math test at random, what is the chance that Tara will be willing to be friends with that person? (3) [52]

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D. If x is binomial, and n  11, find solutions to 1-5 (If you substitute another distribution for the binomial, or any other distribution, justify it!): 1) P3  x  6 when p  .35 . (2)

2) P5  x  8 when p  .65 . (2)

3) P2  x  7 when p  .465 . (3 or 3.5)

1 4) P1  x  3 when p  55 . (2)

5) Px  1 when p  .17 . (2)

6) If x follows the continuous uniform distribution with c  11 and d  17 , find P3  x  14. (2)

7) If x follows the Poisson distribution with a parameter of 1.1 find P3  x  15(2)

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8) If x follows the Poisson distribution with a parameter of 37 find P15  x  45 (2 or 2.5)

9) If x follows the Hypergeometric distribution with N  380 , M  133 and n  11, find P2  x  7 (2) [71]

1 10) (Extra Credit) Find P15  x  45 for an exponential distribution with c  37 , and explain the relation of this problem to 8) above. (3)

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E. Assume that we are considering buying two stocks. For the first of these two stocks  x  1.35 and  x  3.5 . For the second stock  y  2.00 and  y  8.0 . Mark individual questions and parts of questions clearly. (Note: y appears as v in some equations due to a bug in Word formatting: it prints correctly) 1. Assume that the amounts above are in dollars per year and that you buy one share of each stock. Show the mean, standard deviation and coefficient of variation of your return x  y if a)  xy  .8 , b)  xy  .0 and c)  xy  .8 . (8.5) 2. Assume that some time in the future the first stock doubles in value, so that its return is now w  2x and the second stock rises by 50% so that its return is now v  1.5y . Find  v ,  wv ,  wv and  wv if  xy  .8 (4) [83.5] 3. Again  x  1.35 ,  x  3.5 ,  y  2.00 and  y  8.0 . Assume that you have one dollar to invest, so that you will buy P1 shares of stock 1 and P2 shares of stock 2, where P1  P2  1 . Then R  P1 x  P2 y is your return and  x  1.35 ,  x  3.5 ,  y  2.00 and  y  8.0 . Show how  xy affects the minimum risk point, by computing coefficients of variation for R  P1 x  P2 y for P1 equal to 0, .25, . 50, .75 and 1 (You have already done this for zero and one) and for both  xy  .8 and  xy  .8 . This will give you a table with 10 values, some of which are duplicates. Comment on this and from what you learned on the last exam and try to identify ‘no fly’ zones, if there are any. (10+)

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F. (BLK 38) Assume that the amount of gasoline purchased per car at a large service station has an approximately Normal distribution with a mean of $15 and a standard deviation of $4. Find the following. Mark individual questions clearly! 1. The probability that a given car will purchase between $14 and $16 worth of gas. (2) 2. The probability that a random sample of 16 cars will have a sample mean between $14 and $16 (3) 3. The probability that all 16 cars in the sample will purchase an amount between $14 and $16. (2) 4. The probability that at least one of the 16 cars will purchase an amount between $14 and $16. (2) 5. The 95th percentile of the distribution of the sample mean for the 16 cars. (1.5) 6. The 10th percentile of the distribution of the sample mean for 16 cars. (1.5) [95.5]

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ECO 251 QBA1 Name FINAL EXAM, Version 1 Class ______MAY 8, 2006 Student Number ______

Supplementary Correlation Problem

This problem is an edited version of a problem due to Ben-Horim and Levy. (14 points) A textile firm operates in two companies, the United States x and Japan y . Its anticipated profits for 2007 are approximated by the following joint distribution. Please turn in any scratch paper that you use. x  2 2 6 Py  2  .24 .06 0  .30   y 4  .08 .25 .11 .44 12  0 .08 .18 .26   Px .32 .39 .29 1.00

a. Find the expected net profit for both countries and the countries combined. (1.5) b. Find the (population) standard deviation of profits for one or both countries (2) c. Find the (population) covariance and correlation between profits in both countries (3) d. Comment on the strength and the sign of the results and explain from general economic knowledge why you would be very surprised to get a different sign. (1.5) e. What is the standard deviation of the firm’s total profit? (2) f. To verify that the firm’s mean and standard deviation are the values that you presumably got above by using formulas that you learned in class, note that, though there are 9 joint probabilities, x  y can only take 8 values. Fill in the following table and compute Ex  y and  x y using it. If there is a discrepancy, find your error. (4 if they agree)

x  y Px  y x  yPx  y x  y2 Px  y -4 0 2 4 6 10 14 18

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