251x0441 04/26/04 ECO 251 QBA1 Name FINAL EXAM Class ________________ MAY 3, 2004Part I. Do all the Following (14 Points) Make Diagrams! Show your work! Exam is normed on 75 points. There are actually 127 possible points. 3,10~ Nx. 1. 130 xP 2. 91 xP3. 65.14xP4. 32.19F (Cumulative Probability) 5. 4.184.6 xP6. 018.x (Find 018.z first) 7. A symmetrical region around the mean with a probability of 35%.1251x0441 04/26/04 II. (10 points+-2 point penalty for not trying part a .) Show your work! 1x 2x 21x 22x 1 79.2 12.1 6272.64 146.41 2 0.1 1.0 0.01 1.00 3 12.9 -4.2 166.41 17.64 4 16.2 11.0 262.44 121.00 5 23.0 2.5 529.00 6.25 6 22.4 4.4 501.76 19.36 7 64.4 11.8 4147.36 139.24 8 10.2 2.3 104.04 5.29 9 7.6 1.2 57.76 1.44 10 27.7 2.4 767.29 5.76 263.7 44.5 12808.71 463.39The data above represents returns of a sample of 10 low-risk mutual funds in 1999(1x) and the first quarter of 2000(2x). Calculate the following.a. The sample standard deviation1xs of 1x(2) and 2x (1) b. The sample covariance 21xxs between 1x and 2x. (3)c. The sample correlation 21xxr between 1x and 2x. (2)d. Given the size and sign of the correlation, what conclusion might you draw on the relation between 1x and 2x? (1)e. Assume that the return on the funds were .1 higher in the first quarter of 2000. Find,2x 2xs, 21xxs and 21xxr. Use only the values you computed in a-c and rules for functions of x and y to get your results. If you state the results without explaining why, or change 1x and 2x and recompute the results, you will receive no credit. (4).f. Do a 95% confidence interval for the mean return in 2000. (4)g. Was there a significant difference between the return on these funds in 1999 and 2000? A relatively simplistic way to answer this is to check if the mean return in 1999 was in the confidence interval for 2000. (2)[33]2251x0441 04/26/04III. Do at least 4 of the following 6 Problems (at least 12 each) (or do sections 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 xaEaxE 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!!!! 1. Suppose that you pick up 100 packages sent by a mailer. You believe that this mailer produces packageswith a population mean of 6 oz and a population standard deviation of 2.5 oz. Let x represent the sample mean weight that you calculate on this particular day.a. According to the central limit theorem, what distribution, with what mean and standard deviation should apply to the sample mean weight of the packages?(2)b. If you assume that the package weights are normally distributed, what is the probability that a randomly picked package will weigh more than 7 oz? (2)c. What is the probability that the sample mean of the weight of the packages is more than 7 oz? (3) d. What is the probability that the combined weight of all 100 packages is above 40 lbs? (640 oz.) (3)e. If you assume that the package weights are Normally distributed, what is the chance that three randomly picked packages are all above the median weight? (2) [45]f. How heavy does a package have to be to be heavier than 95% of the packages sent out by this mailer? (2) [47]3251x0441 04/26/042. I send out a survey to 200 people. The probability that each person returns the survey is 10%.This is a Binomial problem. a. What is the chance of between 10 and 30 returns. Answer the question by showing that you can use the Normal distribution to solve this problem and solving it.(3)b. Answer the same question by showing that you can use the Poisson distribution to solve it and solving it. (3)c. Let p represent the fraction of the surveys that are returned. Assume that we send out 103 surveys, what is the distribution (including the mean and standard deviation) of p. (2)d. What is the probability that p is above 11% if (i) ,100n(ii) .1000n (4)e. What is the probability that the first survey returned is between the 20th and the 30th sent out? (2) [61]4251x0441 04/26/043. A manufacturer is producing bolts with a nominal length of 5cm. A random sample of 10 bolts is taken from a box containing a large number of bolts. From the sample we get a sample mean of 5.512 and a standard deviation of .22754. a. On the basis of long experience, we know that the standard deviation for the bolts is .2236. Find a 95% confidence interval assuming that this population standard deviation is correct. (2)b. Find a 95% confidence interval for the mean, assuming that the sample standard deviation is correct. (3)c. Find a 95% confidence interval for the mean assuming that the sample standard deviation is correct and that the sample of 10 bolts was taken from a batch of only 50 bolts. (2)d. On the basis of the tests in a-c, is a population mean of 5 cm. reasonable? (1)e. Assume that the median length of the bolts is 5cm., and that all 10 bolts in the sample are more than 5 cm. A p-value is the probability of obtaining a result as extreme or more extreme than what actually happened. If the p-value is below 1% we strongly doubt that the median length is 5cm. An event as extreme or more extreme than what actually occurred is either getting all bolts longer than 5cm or all bolts shorter than 5cm.What is the probability of this happening? (3)f. Assume that a basketball team scores at an average rate of 1.5 points per minute. What is the probability of scoring more than 22 points in a 10 minute interval? What is the probability of scoring less than 10 points in a 10 minute interval? (2)g. By eliminating scores in the bottom 5% or less of the distribution and in the top 5% of the distribution, find an interval between scores with a probability of about
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