Note The enclosed exam solutions are for exam 252x9851 which was given two weeks later than your first exam The questions apply to your exams as follows I Exam 1 and 2 II Exam 1 and 2 III 1 Exam 1 2 Exam 2 3a d Exam 1 3e Exam 2 4 Exam 1 5 Exam 2 6 Exam 1 Computer Problem Exam 2 252y9851 continues in document 252z9851 Exam starts on next page 10 23 98 252y9851 ECO252 QBA2 FIRST HOUR EXAM October 17 1998 Name Show your work I 14 points Do all the following x N 1 5 8 30 1 5 1 1 5 z P 0 06 z 3 56 8 8 P 0 06 z 0 P 0 z 3 56 0239 4998 5237 P 1 x 30 P 1 5 1 5 5 1 5 z P 0 81 z 0 44 2 P 5 x 5 P 8 8 P 0 81 z 0 P 0 z 0 44 2910 1700 4610 9 1 5 3 1 5 z P 0 19 z 0 94 3 P 3 x 9 P 8 8 P 0 z 0 94 P 0 z 0 19 3264 0753 2511 0 1 5 P z 0 19 4 P x 0 P z 8 P 0 19 z 0 P z 0 0753 5000 5753 1 5 1 5 7 1 5 z P 1 06 z 0 3554 5 P 7 0 x 1 5 P 8 8 6 A symmetrical interval about the mean with 63 probability We want two points x 1850 and x 8150 so that P x 1850 x x 8150 6300 From the diagram if we replace x by z P 0 z z 1850 3150 The closest we can come is P 0 z 0 90 3159 So z 1850 0 90 and x z 1850 1 5 0 90 8 1 5 7 2 or 5 7 to 8 7 x 045 7 We want a point x 045 so that P x x 045 045 From the diagram if we replace x by z P 0 z z 045 455 The closest we can come is P 0 z 1 70 4554 So z 045 1 70 and x z 045 1 5 1 70 8 1 5 13 6 or 15 1 2 10 23 98 252y9851 II 6 points 2 point penalty for not trying part a Show your work Samples of five automobiles from one manufacturer and six automobiles from a second manufacturer were tested for top speeds These are listed below Assume that we were sampling from a normal distribution Manufacturer 1 Manufacturer 2 Automobile Speed Automobile Speed 1 110 1 115 2 135 2 130 3 125 3 140 4 185 4 190 5 130 5 135 6 142 Note that x 2 142 s 2 25 4165 a Compute the sample variance s1 of the top speed of automobiles for the first manufacturer Show your work 3 Solution x Speed Manufacturer x2 x 685 1 110 12100 x1 137 n 5 2 135 18225 3 125 15625 2 2 4 185 34225 2 x nx1 137 5 130 97075 516900 s12 Total n 1 685 497075 807 5 or s1 28 4165 b Compute a 99 confidence interval for the mean top speed 1 of automobiles for the first manufacturer 3 Solution From the problem statement 01 From page 10 of the syllabus supplement if the population variance is unknown s1 x t 2 s x and t n 1 t 4005 4 604 2 807 5 28 4165 12 7083 So 1 137 4 604 12 7083 137 58 51 5 n 5 or 76 49 to 195 51 s x1 3 10 23 98 252y9851 III Do at least 3 of the following 6 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 1 Using the data from the previous problem page 2 test the hypothesis that the population mean 1 for the first manufacturer is 135 at the 99 confidence level using a A test ratio 2 b Critical values 2 c A confidence interval 2 d Find an approximate p value for the null hypothesis 1 e Now find a 95 confidence interval for the standard deviation 3 Solution From page 10 of the Syllabus Supplement Interval for Confidence Hypotheses Test Ratio Critical Value Interval Mean known x z 2 x Mean unknown x t 2 s x DF n 1 H 0 0 x 0 x x 0 t sx z H1 0 H 0 0 H 1 0 x cv 0 z 2 x x cv 0 t 2 s x 4 10 135 DF n 1 4 01 t n 1 t 005 4 604 H 0 135 H 1 135 2 From the previous page x1 137 and s x1 s1 n 807 5 28 4165 12 7083 5 5 x 10 137 135 0 1574 This is in the accept region a Test Ratio t s x1 12 7083 between 4 604 and 4 604 so accept H 0 b Critical Value xcv 10 t 2 sx1 135 4 604 12 7083 135 58 51 or 76 5 to 193 5 This includes x1 137 so accept H 0 c Confidence Interval 1 x1 t 2 s x1 137 4 604 12 7083 137 58 51 or 78 5 to 195 5 This includes 10 135 so accept H 0 4 0 134 so that if this were a 1 sided test we could d From the t table t 0 1574 is smaller than t 45 say pvalue 45 Since this is a 2 sided test we double p value and e From page 1 of the Syllabus Supplement n 1 s 2 2 2 n 1 s 2 2 1 2 Since 025 2 n 1 s12 2025 12 and 1 975 look up 2 n 1 s12 2975 or 4 807 5 2 11 1433 1 say 2 pvalue 90 DF n 1 4 and 05 4 4 2025 and 2975 So 4 807 5 0 2158 or 289 86 12 14967 56 Finally taking square roots 17 025 1 122 342 4 10 23 98 252y9851 2 Repeating the data from the previous problem Manufacturer 1 Automobile Speed 1 110 2 135 3 125 4 185 5 130 Manufacturer 2 Automobile Speed 1 115 2 130 3 140 4 190 5 135 6 142 Note that x 2 142 s 2 25 4165 Test the hypothesis that cars produced by the second manufacturer have higher top speeds than those produced by the first manufacturer by comparing means using a A test ratio 3 b Critical values 2 c A confidence interval 2 d Find an approximate p value for the null hypothesis 1 e Now find a 95 confidence interval for the ratio 1 of standard deviations 3 2 Note You may assume either i that population variances are the same for the two parent populations or ii that they are …
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