251x0341 04/21/03 ECO 251 QBA1 Name FINAL EXAM Class ________________ MAY 7, 2003Part I. Do all the Following (14 Points) Make Diagrams! Show your work! 4,3~ Nx. 1. 85.130.2 xP2. 171 xP3. 85.1xP4. 00.4F (Cumulative Probability)5. 00.400.4 xP6. 015.x (Find 015.z first) 7. A symmetrical region around the mean with a probability of 25%Exam is normed on 75 points. There are actually 128 possible points.1251x0341 04/21/03 II. (10 points+-2 point penalty for not trying part a .) Show your work! The following numbers apply to 9 developed countries and give deaths per 100 million miles and speed limits. Row deaths SpLim x y 1 3.1 55 2 3.4 55 3 3.5 55 4 3.6 70 5 4.2 55 6 4.4 60 7 4.8 55 8 5.0 60 9 6.2 75These sums have been calculated for you. 2.38x, 86.1692x, 540yand328502y. Please calculate the following:a. The sample standard deviations of x and y (4) Note that 00.60y and 50.7ys.b. The sample covariance between x and y. (3)c. The sample correlation between x and y. (2)d. Given the size and sign of the correlation, what conclusion might you draw on the relation between speed and safety if this were the only evidence available? (1)e. Assume that the death rate in all 9 countries fell by .1. What would be the new values of ,x ,xsxys and xyr. 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 x and recompute the results, you will receive no credit. (4).2251x0341 04/21/03III. 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!)1. Assume that the amount of paid time (in days) lost by a blue-collar worker during a 3-month period is 3.1,4.1N. I take a random sample of 10 workers and record the time they lost in the last 3 months..a. What is the probability that a randomly picked worker lost paid time exceeding 1.5 days in the 3-month period? (2)b. What is the probability that all 10 workers in the sample lost paid time exceeding 1.5 days in the 3-month period? (2)c. What is the probability that at least one of the workers in the sample lost paid time exceeding 1.5 days in the 3-month period? (2)d. What is the probability that the average amount of paid time lost time exceeded 1.5 days in thethree month period? (2)e. What is the probability that the total amount of time lost by the sample of 10 workers exceeded 15 days in the three month period. (2)f. Looking at the distribution of the sample mean in this problem, give a value of the sample mean that will be above the mean we actually observe 95% of the time (the 95th percentile) (2) 3251x0341 04/21/032. (Bowerman and O’Connell) A retailer that sells home entertainment systems accumulated 10,451 sales invoices during the last year. a. An auditor takes a sample of 16 invoices and computes mean sales of .532$xIf the population standard deviation was known to be $168, find a 99% confidence interval for the mean sales per invoice. (4)b. I lied. Though the sample mean was $532, $168 was a sample standard deviation. Do the 99% confidence interval again. (4)c. I lied. Though it is true that the sample mean was $532 and the sample standard deviation was $168, the actual sample was 650 invoices out of the 10451 invoices that were collected. Do the 99% confidence interval again. (4)d. (Extra credit) Assume that the confidence interval in c is correct, and that the 10451 invoices were all that were generated, using these two facts, create a confidence interval for total sales in the last year. (3)e. (Extra credit) The firm claims that its total sales were above $5.75 million last year. In view of your results in d, does that seem likely? Would you change your mind if I insisted on a confidence level of 99.8%? (3)f. Using the data in a) create a 97% Confidence interval for the mean sales per invoice. (You might want to look at page 1.) (2)4251x0341 04/21/033. a. Assume that the entire amount of a product made by a supplier is a population of 100 units and that you buy the whole batch. Assume that 15% of the batch is defective. Take a sample of 10 items and give me the probability that at least one is defective. (3)b. Assume that the batch you buy is much larger, well over 200 and that you still take a sample of 10. What is the chance that at least one is defective? (You should not need to use the number 200 or any largernumber in your calculations.) (2)c. Assume that you have bought at least a million units, and that 15% still represents the proportion of the product that is defective. This time you take a sample of 80. Find the probability that at least 10 are defective using the Poisson distribution. First, show that it is legitimate to use the Poisson distribution in this case. (3)d. Do part c using the Normal distribution. That is assume that you have a large population that is 15% defective and that you take a sample of 80. Show that the Normal distribution can be used here and find the probability of at least 10 defective items in the sample. (3 points without continuity correction, 3.5 with)e. (Extra credit) In section 7.3 of the text (on the CD) the author tells us we should use a finite population correction with the variance if it is justified. Assume that in part d, the population is 200 and we take a sample of 80, what is the probability of at least 10 defective items in the sample now? (2)f. If we are taking a sample from a large population, find the probability that the first defective item is between the 6th to the 10th item we test. (2) 5251x0341 04/21/034. As everyone knows, a jorcillator has two components, a Phillinx and a Flubberall. It seems that the jorcillator only
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