12/26/99 251y9941 ECO251 QBA1 Name key FINAL EXAM DECEMBER 14, 1999Part I. Do all the Following (20 Points) Make Diagrams!A. z N~ ( , )0 11. 43.1zP 9236.4236.5.43.100 zPzP Diagram! These are expected and will be provided in class for version 2.2. 24.343.1 zP 0758.4236.4994.43.1024.30 zPzP3. 43.143.1 zP 8472.4236.243.102 zP4. 65.095.0 zP 0867.2422.3289.065.0095.0 zPzP5. 97.z This is the point with a probability of .97 above it or .03 below it. It is the 3rd percentile of .z So 03.97.zz . From the diagram .4700.003. zzP the closest we can come to this probability using the Normal table is .4699.88.10 zP So.88.103.97. zzNote that probabilities cannot be above 1 or below 0.112/26/99 251y9941B. 3,5.1~ Nx.As usual, people made diagrams of x with zero in the middle. Make up your mind! If you are diagramming x, put the mean in the middle; if you are diagramming z put zero in the middle.1. 43.1xP 4920.0080.5.002.0002.035.143.1 zPzPzPzP Remember xz.2. 24.343.1 xP 58.002.035.124.335.143.1 zPzP 2270.2190.0080.58.00002.0 zPzP3. 43.143.1 xP 02.098.035.143.135.143.1 zPzP 3285.0080.3365.002.0098.0 zPzP4. 65.095.0 xP 72.082.035.165.035.195.0 zPzP 0297.2642.2939.072.0082.0 zPzP 5. 97.x This is the point with a probability of .97 above it or .03 below it. It is the 3rd percentile of x. Because it is below the 50th percentile, which is the mean for the Normal distribution, the point we want will be below 1.5. On the previous page we found .88.103.97. zz So97.97.zx .14.4388.15.1 To check to see if this is correct: 14.4xP 88.135.114.4 zPzP 97.9699.4699.5.088.10 zPzP212/26/99 251y9941II. (4 points-2 point penalty for not trying .) Show your work! A manufacturer of printheads investigates the amount of time before a printhead fails. The data for a sample of 8(in millions of characters)are below. x 1.5 1.4 1.2 1.6 1.3 1.3 1.4 1.5 Compute the sample standard deviation, s. Show your work. (4) Solution: Printhead lives x 2x 1 1.5 2.25 2 1.4 1.96 3 1.2 1.44 4 1.6 2.56 5 1.3 1.69 6 1.3 1.697 1.4 1.968 1.5 2.25 11.2 15.80 40.182.11nxx 017143.0712.0740.1880.1512222nxnxsx13093.0017143.0 xs 046291.0813093.0nssxx312/26/99 251y9941III. Do at least 4 of the following 7 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!1 1. Using the sample standard deviation from the sample of 8 on the previous page2 a. Compute a 95% confidence interval for the mean time before a printhead fails. (4)b. Compute a 95% confidence interval for the mean time before a printhead fails assuming that the sample of 8 was taken from a batch of 80 printheads. (4)c. Compute a 95% confidence interval for the mean time before a printhead fails assuming that the population standard deviation is known to be 1.40 and the population is very large, but the sample size is still 8. (4)Solution: From the previous page 40.1x,017143.02xs, 13093.0xs,046291.0nssx.a) For a) and b) population variance is unknown, so we must use t instead of z. The general formula for a confidence interval is xnstx12. Here the degrees of freedom are 7181 nand since the confidence level is 95%, 95.1 or 05.. Also 365.27025.12ttn.From the previous problem - 13093.0017143.0 xs 046291.0813093.0nssxx Putting this all together xnstx12 1095.040.1046291.0365.240.1 or 1.29 to 1.51. More formally, 95.51.129.1 Pb) The population size, 80N, is less than 20 times the sample size 8nso 1NnNnssxx 04419.09546687.0046291.0180880813093.0and xnstx12 1045.04.104419.0365.24.1 or 1.30 to 1.50.c) This time 40.1xis known, so we can replace twith 960.1025.2zz.4950.0840.1nxx and xzx2 9702.04.14950.0960.14.1 or 0.43 to 2.37.412/26/99 251y99412. The Freshman class at Chest Wester University has SAT Math scores that are normally distributed witha mean of 480 and a standard deviation of 60. a. What is the probability that a class of 36 will have a mean score below 476? (3)b. What is the probability that a person randomly picked from the Freshman class will have a score below 476? (2)c. Assuming that the class is quite large, what is the probability that 10 people picked randomly from the Freshman class will all have a score below 476? (2)d. What score would you have to have to be at the 95th percentile of this class? (3)e. Create a symmetrical interval around 480 that contains 80% of the Freshman math scores. (2).f. What is the probability that, if I pick 10 people from the Freshman class, 8 or more of them will have scores above 480? (2) Solution: From the problem statement 60,480~ Nx, i.e. 60,480 and for the samplemean, .103660nxxa) 3446.1554.5.040.0040.010480476476 zPzPzPzPxPb) 4721.0279.5.007.0007.060480476476 zPzPzPzPxPc) Binomial probability of 10 out of 10 when 4721.p is .00054997.104997.54721.410d) From the t table 645.105.z. So
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