252y0552 10 31 05 Open in Print Layout format ECO252 QBA2 FIRST HOUR EXAM October 17 18 2005 Name Key Hour of class registered Class attended if different Show your work Make Diagrams Exam is normed on 50 points Answers without reasons are not usually acceptable I 8 points Do all the following If you do not use the standard Normal table explain x N 4 5 6 Make diagrams For x draw a Normal curve with a vertical line in the center at 4 5 For z draw a Normal curve with a vertical line in the center at zero 3 4 5 17 5 4 5 P 3 67 z 0 25 z 1 P 17 5 x 3 P 6 6 P 3 67 z 0 P 0 25 z 0 4999 0987 4012 1 252y0552 10 31 05 Open in Print Layout format 7 4 5 4 5 4 5 z P 0 z 0 42 1628 2 P 4 5 x 7 00 P 6 6 10 22 4 5 3 P x 10 22 P z P z 0 95 6 P z 0 P 0 z 0 95 5 3289 1711 2 252y0552 10 31 05 Open in Print Layout format 4 x 06 There is no excuse for not being able to do this at this point since you should have done it 3 times in the take home To find z 06 make a Normal diagram for z showing a mean at 0 and 50 above 0 divided into 6 above z 06 and 44 below z 04 So P 0 z z 06 4400 The closest we can come is P 0 z 1 55 4394 or P 0 z 1 56 4406 So use z 06 1 555 x 06 z 06 4 5 1 555 6 13 83 3 252y0552 10 31 05 Open in Print Layout format II 5 points 2 point penalty for not trying part a A dealer wishes to test the manufacture s claim that the Toyota Caramba gets 28 miles per gallon The data below is found Recomputing what I ve done for you is a great way to waste time a Compute the sample standard deviation s of the miles per gallon Show your work 2 b Is the population mean significantly different using a 95 confidence level from 28 mpg Show your work 3 c Extra Credit Find an approximate p value for your null hypothesis 2 Solution a Compute the sample standard Row x x2 deviation s of the miles per gallon Show your 1 23 56 555 074 work 2 2 26 09 680 688 3 27 55 759 003 x 263 98 x 26 398 4 26 92 724 686 n 10 5 29 20 852 640 x 2 nx 2 6 29 02 842 160 7010 07 10 26 398 2 2 s 7 22 48 505 350 n 1 9 8 27 45 753 503 41 52596 9 25 90 670 810 4 61400 9 10 25 81 666 156 sum 263 98 7010 070 s 4 61400 2 148021 Note that excessive rounding can throw this answer way off Using x 26 I got 7010 10 26 2 7010 6760 250 27 778 and s 5 3541 9 9 9 b Is the population mean significantly different using a 95 confidence level from 28 mpg Show your work 3 Given x 26 398 s 2 14802 n 7 and 05 DF n 1 9 and 9 t 025 2 262 s2 We are testing H 0 28 Use only one of the following Confidence Interval s x s n 2 14802 10 4 61400 0 461400 0 6793 10 x t 2 s x 26 398 2 262 0 6793 26 398 1 5366 or 21 861 to 27 935 Since 28 is not on the confidence interval it is significantly different from the sample mean s 2 14802 4 61400 0 461400 0 6793 Critical Value s x 10 n 10 x cv t 2 s x 28 2 262 0 6793 28 1 5366 or 26 46 to 29 53 Since 26 398 is not between the critical values it is significantly different from the population mean s 2 14802 4 61400 0 461400 0 6793 Test Ratio s x 10 n 10 t x 0 26 398 28 2 358 The accept zone is between 2 262 and 2 262 Since sx 0 6793 2 197 is not between these values reject the null hypothesis c Extra Credit Find an approximate p value for your null hypothesis 2 9 9 2 358 is between t 025 2 262 and t 01 2 821 Since pval 2 P t 2 358 it is between 02 and 05 4 252y0552 10 31 05 Open in Print Layout format III Do all of the following problems 2 each unless marked otherwise adding to 18 points Show your work except in multiple choice questions Actually it doesn t hurt there either If the answer is None of the above put in the correct answer 10 1 For sample sizes greater than 100 the sampling distribution of the mean will be approximately normally distributed a Regardless of the shape of the population b Only if the shape of the population is symmetrical c Only if the standard deviation of the samples are known d Only if the population is normally distributed 2 When determining the sample size for a proportion for a given level of confidence and sampling error the closer to 0 50 that p is estimated to be the the sample size required a Smaller b Larger c Sample size is not affected d The effect cannot be determined from the information given 3 Which of the following would be an appropriate null hypothesis a The population proportion is less than 0 65 b The sample proportion is less than 0 65 c The population proportion is at most 0 65 d The sample proportion is at most 0 65 4 A Type I error is committed when a We reject a null hypothesis that is true b We don t reject a null hypothesis that is true Note that b and c are not errors c We reject a null hypothesis that is false d We don t reject a null hypothesis that is false 5 If we are performing a two tailed test of whether 100 the power of the test in detecting a shift of the mean to 105 will be its power detecting a shift of the mean to 96 a Less than 96 is closer than 105 to 100 b Greater than c Equal to d Not comparable to 6 From a sample of 100 students we find that the mean expenditure for books is 316 40 with a sample standard deviation of 43 20 You are asked to test whether the population mean expenditure is above 314 using a 10 significance level a What are the null and alternative hypotheses 2 b What is the rejection zone in terms of x 2 c What is your conclusion 2 16 …
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