3 22 00 252y0021 ECO252 QBA2 SECOND HOUR EXAM March 21 2000 Name key Hour of class registered Class attended if different Show your work Make Diagrams I 14 points Do all the following x N 9 6 9 9 1 9 z P 1 67 z 0 4525 1 P 1 x 9 P 6 6 11 9 11 9 z P 3 33 z 0 33 2 P 11 x 11 P 6 6 P 3 33 z 0 P 0 z 0 33 4996 1293 6289 0 9 9 9 z P 3 00 z 1 50 3 P 9 x 0 P 6 6 P 3 00 z 0 P 1 50 z 0 49865 4332 06545 15 9 4 F 15 The cumulative probability up to 15 P x 15 P z 6 P z 1 00 P z 0 P 0 z 1 00 5 3413 8413 17 5 9 14 9 z 5 P 14 x 17 5 P P 0 83 z 1 42 6 6 P 0 z 1 42 P 0 z 0 83 4222 2967 1255 6 A symmetrical interval about the mean with 48 probability We want two points x 74 and x 26 so that P x 74 x x 26 4800 From the diagram if we replace x by z P 0 z z 26 2400 The closest we can come is P 0 z 0 64 2389 or P 0 z 0 65 2422 Since these are about the same distance from 2400 use z 26 0 645 and x z 26 9 0 645 6 9 3 87 12 87 9 5 13 9 z or 5 13 to 12 87 To check this note that P 5 13 x 12 87 P 6 6 P 0 645 z 0 645 2 P 0 z 0 645 2 2400 4800 7 x 085 We want a point x 085 so that P x x 085 085 From the diagram if we replace x by z P 0 z z 085 4150 The closest we can come is P 0 z 1 37 4147 So z 085 1 37 and x z 085 9 1 37 6 9 8 22 or 17 22 17 22 9 P z 1 37 To check this note that P x 17 22 P z 6 P z 0 P 0 z 1 37 5 4147 0853 085 3 22 00 252y0021 II 6 points 2 point penalty for not trying A manufacturer of mowers wants to compare the amount of time it takes to mount a mower engine using two different processes Two independent samples are taken and the time in minutes appears below Note Most of you didn t bother to tell me what your hypotheses were in any problem but the first one This means that I couldn t really tell whether your critical values etc were right or wrong x1 x2 2 1 4 1 9 1 3 1 2 1 3 3 7 3 5 3 8 3 4 3 3 3 s 2 1 Note that x1 4 1 and s1 2 915476 8 5 a Compute the sample standard deviation for the second process s 2 Show your work 3 b Compute a 95 confidence interval for the difference between the population means 1 and 2 assuming that the variances are equal for the two parent populations According to your confidence interval is there a significant difference between the population means You must tell why 3 a Solution x 2 5 3 and s 2 2 097618 x2 x 22 3 3 7 3 5 3 10 89 53 29 28 09 s 2 2 4 4 b From page 10 of the Syllabus Supplement Interval for Confidence Hypotheses Interval Difference H 0 0 d t 2 sd between Two x2 H 1 0 31 8 Means x 2 5 s 30 1 1 s 1 2 p 6 d unknown n n1 n 2 variances DF n1 n 2 2 2 assumed xequal 2 nx 2 s22 2 2 190 54 6 5 30 5 8 3 4 3 3 3 31 8 68 89 18 49 10 89 190 54 Test Ratio t Critical Value d 0 sd s 2p d cv 0 t 2 sd n1 1 s12 n2 1 s22 n1 n 2 2 n 1 4 40 s 2 2 097618 x1 4 1 s12 8 50 s1 2 915476 x 2 5 3 s 22 4 40 s 2 2 097618 d x1 x 2 4 1 5 3 1 2 s 2p n1 1 s12 n2 1 s22 sd s p n1 n2 2 1 1 n1 n 2 DF n1 n 2 2 5 6 2 9 4 8 50 5 4 40 34 00 22 00 6 22222 9 9 6 22222 1 1 5 6 6 22222 366667 05 9 t 025 2 262 2 28148 1 51046 2 3 22 00 252y0021 Confidence Interval d t 2 sd 1 20 2 262 1 51046 1 20 3 417 or 4 62 to 2 22 The interval includes 0 so there is no significant difference between the means H 0 0 H 0 1 2 H 0 1 2 0 Formally our hypotheses are H 0 or or We do not reject 1 H1 1 2 H1 1 2 0 1 2 H0 3 3 22 00 252y0021 III Do at least 3 of the following 4 Problems at least 10 each or do sections adding to at least 30 points Anything extra you do helps and grades wrap around Show your work State H 0 and H 1 where appropriate Use a 95 confidence level unless another level is specified 1 For your convenience the data from the previous page giving samples of the time to mount a mower engine is copied below Test the hypothesis that the mean time to mount a mower is greater for method two than method one x1 x2 2 1 4 1 9 1 3 1 2 1 3 3 7 3 5 3 8 3 4 3 3 3 Note that x1 4 1 and s1 2 915476 s 2 1 8 5 a Choose the correct null hypothesis from the list below and write the alternative hypothesis 2 H 0 1 2 H 0 1 2 H 0 1 2 H 0 1 2 H 0 1 2 H 0 x1 x 2 H 0 x1 x 2 H 0 x1 x 2 H 0 x1 x 2 H 0 x1 x 2 H0 2 H0 2 H0 2 H 0 2 H 0 2 4 1 4 1 4 1 4 1 4 1 b Assume that 1 2 and find a critical value appropriate for this problem using a confidence level of 90 3 c Use your critical value to test the hypothesis State clearly whether you reject the null hypothesis 2 d Repeat the test using i a test ratio 2 and ii a confidence interval 2 find an approximate p value for your null hypothesis 1 e Do a …
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