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WCU ECO 251 - ECO 251 Third Exam

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251x0531 4/18/05 ECO251 QBA1 THIRD EXAMApr 25, 2005 Name: _____________________ Student Number: _____________________ Class Time (Circle) 1pm 2pmPart I: 16 points.z follows the standardized Normal distribution   1,0~ Nz. Find the following. Make diagrams! 1.  63.263.2  zP2.  63.211.3  zP3.    11.311.3  zPF4. 115.z251x0531 4/18/05x follows the Normal distribution   2,63.2~ Nx. Find the following. Make diagrams! 5.  63.263.2  xP6.  63.211.3  xP7.    11.311.3  xPF8. 115.x2251x0531 4/18/05Part II: (15+ points) Do all the following: All questions are 2 points each except as marked. Exam is normed on 50 points including take-home. Most questions come from Wonnacott and Wonnacott (1990) (Showing your work can give partial credit on some problems! In open-ended questions it is expected. Please indicate clearly what sections of the problem you are answering and what formulas you are using. Neatness counts!) Remember that you may not be able to finish this section, so ration your time on each problem. [Numbers in brackets are a cumulative total]1. If y tends to decrease as x increases, what can we say about the population correlation ? a)1.b)1.c)0.d)0.2. The riskiness of a portfolio made up of two investments a) will be higher when the covariance is zero.b) will be higher when the covariance is negative.c) will be higher when the covariance is positive.d) does not depend on the covariance. [2]3. Seventy items are randomly selected from a pilot production run of N items to check their quality. x is the number of defective items in the sample. The distribution of x can be considered approximately binomial if N is a) 15b) 152c) 400d) 1500e) All of the abovef) None of the above.4. There are 24 Million people living in California. Approximately 8 million are in Los Angeles. The Census Bureau takes a random sample of 200 people from the state. Let x be the number of people in the sample who live in Los Angeles, then x is most conveniently treated as having the a) Binomial distribution with millionn 24 and 248p.b) Poisson distribution.c) Binomial distribution with 200n and 248p.d) Hypergeometric distributione) none of the above.5. Let x represent the number of times a fair die (6 sides) comes up with a 1 on top when the die is cast 300 times. What is a)  xE, b) ),(xVar c)  2xE Show your work! (5) [11] 6. Assume that a fair 6-sided die is cast a large number of times. What is the chance that the first timeit comes up with a 1 on top is in the first 10 throws? Show your work! (2)3251x0531 4/18/057. A student calculates that      935359 FFxP . The student could be working with the a) Binomial Distributionb) Poisson Distribution.c) Continuous Uniform Distributiond) Geometric distributione) All of the abovef) None of the above. [15]8. If we are taking a sample of 3 from a population that is one-third defective, what is the probability that at least one item in our sample is defective if a) 30N? and b) N is very large? Show your work! (4)9. (Extra credit) The time it takes to complete an exam has an exponential distribution with a mean of 40 minutes. Luce Cash is in a hurry to finish so she can go shopping. What is the probability that she will finish the exam in 30 minutes or less? What is the probability that she will still be working when the instructor collects the exams after 55 minutes? 4251x0531 4/18/05 ECO251 QBA1 THIRD EXAMApr 25, 2005TAKE HOME SECTION- Name: _________________________ Student Number: _________________________Throughout this exam show your work! Please indicate clearly what sections of the problem you are answering and what formulas you are using.Part III. Do all the Following (19+ Points) Show your work! 1. Before you start, personalize the data below as follows. Take the third to last digit of your student number and add it to the last digit of .06. Take half the third to last digit of your social security number and subtract it from each of the .09s. A photographic processor is trying to determine the distribution of the times that the business can promise to customers. xis the time in days it takes the package to arrive and y is the time in days it takes the package to return to the customer. It takes 2 days to process the film, so that the total time is 2 yxt days.x 2 3 41 .10 .15 .05y2 .15 .26 .093 .05 .09 .06For example Robin dePoore’s student number is 999799 so she changes the .06 to .06 + .07 = .13 and the two .09s to .09 - .035 = .055. This should not change the total sum of the numbers in the joint probability table. Find.a) The mean and standard deviation of both x andy. (2)b) The covariance and correlation between x andy. (3)c) Using only the results in a) and b), find the mean and variance of t (2). d) Find the probability that the total time 2 yxt is no more than 5 days (2).e) Assume that because of an anthrax scare,x, the time it takes the package to arrive, doubles. Using only the numbers that you found in a) and b) and appropriate formulas, find the new covariance and correlation and the mean and variance of t (2).f) Go back to the original values of x and y and assume that the marginal probabilities of x andy are correct, what would the joint probability table look like if x andywere independent? (1)g) Recall what the covariance of x andywould be if they were independent. What would the variance of 2 yxt be now? (1) [13]2. Personalize the data below by adding the third to last digit of your student number to 10.x y23 816 1320 921 10a) Using the computational formula, find the sample variance of x . (2)b) Find the covariance and correlation between x and y.(2)c) Find the covariance and correlation between 32  xw and.97  yvUse only what you found in b) and appropriate formulas.d) If 32  xw and ,97  xvfind the covariance and correlation between wand v. Use only what you found in b) and appropriate formulas. (Hint: figure out or find from problems what  xxCov


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