251y0641 5/10/06ECO 251 QBA1 Name KEY FINAL EXAM, Version 1 Class ________________ MAY 8, 2006Part 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). 6.5,13~ Nx 1. 280 xP 68.232.26.513286.5130 zPzP 68.20032.2 zPzP 9861.4963.4898.. For 6.5,13~ Nxmake a Normal curve centered at 13 and shade the area from 0 to 28; for z make a Normal curve centered at zero and shade the area from -2.32 to 2.68. Since the diagrams show anarea on both sides of the mean, you add.2. 00.12F (Cumulative Probability) 12xP 18.06.51312 zPzP 018.00 zPzP=.5-.0714=.4286For 6.5,13~ Nxmake a Normal curve centered at 13 and shade the area below 12; for z make a Normal curve centered at zero and shade the area below -0.18. Since the diagrams show an area below the mean that does not touch the mean, you subtract.3. 28xP 68.26.51328 zPzP 68.200 zPzP0037.4963.5. For 6.5,13~ Nxmake a Normal curve centered at 13 and shade the area above 28; for z make a Normal curve centered at zero and shade the area above 2.68. Since the diagrams show an area above the mean that does not touch the mean, you subtract4. 3228 xP 39.368.26.513326.51328 zPzP 68.2039.30 zPzP0034.4963.4997. For 6.5,13~ Nxmake a Normal curve centered at 13 and shade the area above 28; for z make a Normal curve centered at zero and shade the area between 3.39 and 2.68. Since the diagrams show an area above the mean that does not touch the mean, you subtract5. 33 xP 79.168.26.51336.5133 zPzP 079.1068.2 zPzP0300.4663.4963. For 6.5,13~ Nxmake a Normal curve centered at 13 and shade the area between -3 and 3, both ofwhich are below 13; for z make a Normal curve centered at zero and shade the area between -2.68 and -1.79. Since the diagrams show an area below the mean that does not touch the mean, you subtract. 1251y0641 5/10/06 6.5,13~ Nx6. 23.x(Find 23.z first) . Solution: Make a diagram. The diagram for z will show an area with a probability of 100 - 23% = 77%. below 23.z . The area below 23.z is split by a vertical line at zero into two areas. The lower one has a probability of 50% and the upper one a probability of 77% - 50% = 27%. The upper tail of the distribution above 23.z has a probability of 23%, so that the entire area above 0 adds to 27% + 23% = 50%. From the diagram, we want one point 23.z so that 77.23zzP or 2700.029 zzP. If we try to find this point on the Normal table, the closest we can come is 2704.74.00 zP, but 2673.73.00 zP not as close, but is acceptable. So74.023.zSince 6.5,13~ Nx, the diagram for x would show 77% probability split in two regionson either side of 13 with probabilities of 50% below 13 and 27% above 13 and below 23.x, and with 23% above 23.x. 74.023.z, so the value of xcan then be written 23.23.zx .144.17144.4136.574.013 To check this: 144.17xP 74.06.513144.17 zPzP 74.000 zPzP.23.2296.2704.5000. 7. A symmetrical region around the mean with a probability of 23%. [14, 14]Solution: Make a diagram. The diagram for z will show a central area with a probability of 23%. It is split in two by a vertical line at zero into two areas with probabilities of 11.5%. The tails of the distribution each have a probability of 50% - 11.5% = 38.5%. From the diagram, we want two points385.z and 615.z so that 2300.385.615. zzzP. The upper point, 385.z will have 1150.2%230385. zzP, and by symmetry 385.615.zz . From the interior of the Normal table the closest we can come to .1150 is 1141.29.00 zP, which is slightly too low. The next best point would be 0.30 since 1179.30.00 zP. We can say 29.0385.z, and our 23% symmetrical interval for z is -0.29 to 0.29. Since 6.5,13~ Nx, the diagram for x (if we bother) will show 23% probability split in two 11.5% regions on either side of 13, with 38.5% above 385.x and 38.5% below 855.x. The interval for x can then be written 385.zx 624.1136.529.013 or 11.376 to 14.624. To check this: 624.14376.11 xP 29.029.06.513624.146.513376.11 zPzP %232281.1141.229.002 zP 2251y0641 5/10/06II. (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 71.46In order to speed things up, I have computed the sum of the first seven observations. 71ix610.000, 71iy501.950, 712ix63720.1, 712iy35996.0 and 71ixy43607.4.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 ys of average score. (2)c. The sample covariance xys between x and y. (2)d. The sample correlation xyr 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 (xw 85.). Find w (the sample mean of earnings), 2ws, wys and wyr. Use only the
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